1. Introduction
In this paper, we are interested in local behaviour of the weak solutions to nonlocal double phase problem in the Heisenberg group ${{\mathbb{H}}^n}$, whose prototype is

where $1 \lt p\le q \lt \infty$,
$s,t\in(0,1)$,
$ a(\cdot,\cdot)\ge 0$,
$Q=2n+2$ is the homogeneous dimension and Ω is an open bounded subset of
$\mathbb{H}^n$ (
$n\ge1$). In the display above,
$\|\cdot\|_{\mathbb{H}^n}$ and
$\mathrm{P.V.}$ mean the standard Heisenberg norm and “in the principal value sense”, respectively. The main feature of the integro-differential equation (1.1) is that the leading operator could change between two different fractional elliptic phases according to whether the modulating coefficient a is zero or not.
We observe that, if the coefficient $a\equiv0$, equation (1.1) is reduced to the p-fractional subLaplace equation arising in many diverse contexts, such as quantum mechanics, image segmentation models, ferromagnetic analysis and so on. Let us pay attention to the linear scenario first, i.e., p = 2. This kind of problems can be regarded as an extension of the conformally invariant fractional subLaplacian
$\left(-\Delta_{\mathbb{H}^n}\right)^s$ in
$\mathbb{H}^n$ proposed initially in [Reference Branson, Fontana and Morpurgo2] by the spectral formula

where $s\in(0,1)$,
$\Gamma(\cdot)$ is the Euler Gamma function, T is the vertical vector field, and
$\Delta_{\mathbb{H}^n}$ is the typical Kohn–Spencer subLaplacian on
$\mathbb{H}^n$. Subsequently, Roncal and Thangavelu [Reference Roncal and Thangavelu36] demonstrated the representation as below

holds true for $C(n,s) \gt 0$ depending only on
$n,s$. During the last decade, several aspects of the fractional operator of the type (1.2) have been investigated, such as Hardy and uncertainty inequalities on stratified Lie groups [Reference Ciatti, Cowling and Ricci6], Sobolev and Morrey-type embedding theory for fractional Sobolev space
$H^s(\mathbb{H}^n)$ [Reference Adimurthi and Mallick1], Harnack and Hölder estimates in Carnot groups [Reference Ferrari and Franchi18], Liouville-type theorem [Reference Cinti and Tan7]. One can refer to [Reference Ferrari, Miranda, Pallara, Pinamonti and Sire19–Reference Garofalo and Tralli22] and references therein for more results on the linear case. Regarding the nonlinear analogue to (1.2), the p-growth scenario is considered (
$p\not=2$). For what concerns the regularity properties of weak solutions to the fractional p-subLaplace equations on the Heisenberg group, Manfredini et al. [Reference Manfredini, Palatucci, Piccinini and Polidoro31] established the interior boundedness and Hölder continuity via employing the De Giorge–Nash–Moser iteration; see also [Reference Palatucci and Piccinini32] for the nonlocal Harnack inequality, where the asymptotic behaviour of fractional linear operator was proved as well. In addition, as for the obstacle problems connected with the nonlocal p-subLaplacian, we refer to [Reference Piccinini34] in which Piccinini studied systematically solvability, semicontinuity, boundedness and Hölder regularity up to the boundary for weak solutions. More interesting estimates or fundamental functional inequalities can be found in [Reference Kassymov and Suragan27, Reference Kassymov and Surgan28, Reference Palatucci and Piccinini33]. To some extent, we can see that the results mentioned above extended the counterparts of the fractional Euclidean setting in [Reference Di Castro, Kuusi and Palatucci13, Reference Di Castro, Kuusi and Palatucci14, Reference Iannizzotto, Mosconi and Squassina26, Reference Korvenpää, Kuusi and Palatucci29, Reference Korvenpää, Kuusi and Palatucci30] to the Heisenberg framework.
Equation (1.1) could be viewed naturally as the nonlocal version of the classical double phase problem of the following type

Within the Euclidean context, the regularity theory of weak solutions to (1.3) or minimizers of the corresponding functionals has been developed extensively, beginning with the pioneering papers of Colombo and Mingione [Reference Colombo and Mingione8, Reference Colombo and Mingione9]. Under $a\in L^\infty_{\rm loc}(\Omega)$ and,
$p\le q\le \frac{np}{n-p}$ if p < n, or
$p\le q \lt \infty$ if
$p\ge n$, local boundedness for u was shown; and further under
$u\in L^\infty_{\rm loc}(\Omega)$,
$a\in C^{0,\alpha}_{\rm loc}(\Omega)$ and
$p\le q\le p+\alpha$, Hölder continuity of u was obtained as well, see, e.g. [Reference Colombo and Mingione9, Reference Cupini, Marcellini and Mascolo10].
Very recently, the investigation of nonlocal problems with nonstandard growth, especially of those with (p, q)-growth condition, has been attracting increasing attention, however only in the fractional Euclidean spaces. In this respect, De Filippis and Palatucci [Reference De Filippis and Palatucci12] introduced nonlocal double phase equations of the form (1.1) in the Euclidean spaces, and established Hölder continuity for bounded viscosity solutions. Weak theory on this class of nonlocal equations was rapidly explored in hot pursuit, for example, [Reference Scott and Mengesha37] for self-improving inequalities on bounded weak solutions, [Reference Fang and Zhang17] for Hölder regularity and relationship between weak and viscosity solutions in the differentiability exponents $s\ge t$, [Reference Byun, Ok and Song4] for Hölder property with weaker assumption on solutions in the case s < t, [Reference Giacomoni, Kumar and Sreenadh24] for the sharp Hölder index and the parabolic version. Concerning more regularity and related results for nonlocal problems possessing nonuniform growth, one can see [Reference Byun, Kim and Ok3, Reference Chaker, Kim and Weidner5, Reference Fang and Zhang16, Reference Giacomoni, Kumar and Sreenadh23, Reference Prasad and Tewary35] and references therein.
In particular, we would like to mention that Palatucci, Piccinini, et al. in a series of papers [Reference Manfredini, Palatucci, Piccinini and Polidoro31–Reference Palatucci and Piccinini33] proposed the open problems about the regualrity of solutions to the so-called nonlocal double phase equation in the Heisenberg group $\mathbb{H}^n$. In this paper, influenced by the works [Reference Byun, Ok and Song4, Reference Di Castro, Kuusi and Palatucci14] we answer this question and develop the local regularity theory for the weak solutions of such equations in the Heisenberg group
$\mathbb{H}^n$, including the boundedness and Hölder continuity of solutions. The main difficulties which are different from the previous ones are mainly two parts. One is that equation (1.1) not only possesses the nonlocal feature of the embraced integro-differential operators and the noneuclidean geometrical structure of the Heisenberg group, but also inherits the typical characteristics exhibited by the (local) double phase problems due to the (p, q)-growth condition and the presence of the nonnegative variable coefficient a. We need to find some appropriate assumptions on the summability exponents
$p,q\in (1, \infty)$ and differentiability exponents
$s,t\in (0,1)$ together with the variable coefficient a in order to locally rebalance the non-uniform ellipticity of the operator. The other one is that the existing Sobolev embedding theorem, lemma 2.2, cannot be applied to our setting directly. To overcome this point, we have to establish a suitable Sobolev–Poincaré type inequality on balls in the Heisenberg group
$\mathbb{H}^n$. It may be of independent interest and potential applications when investigating regularity properties for some other nonlocal equations in the Heisenberg group. These difficulties make the current study more challenging than the fractional p-subLaplacian case.
Now we are in a position to state our main contributions. We first collect some notations, definitions as well as assumptions. Let $s,\;t$ and
$p,\;q$ satisfy

and the coefficient $a:{\mathbb{H}^n}\times {\mathbb{H}^n} \to {\mathbb{R}^+}$ fulfil

and

for $\left( {\xi ,\eta } \right),\left( {\xi ',\eta '} \right) \in {\mathbb{H}^n} \times {\mathbb{H}^n}$ and
$\alpha \in \left( {0,1} \right]$.
For convenience, we introduce the following notations:

and

with ${\tau _1},{\tau _2} \in \mathbb{R}$ and
$l \in \{p,q\}$, and

for every measurable set $\Omega \subset {\mathbb{H}^n}$ and
$u: \Omega \to \mathbb{R}$. A function space related to weak solutions to (1.1) is defined as

where

Additionally, in view of the nonlocal nature of this problem, we need define a tail space

and the nonlocal tail

We can notice that the quantity T is finite if $u\in L^{q-1}_{sp}(\mathbb{H}^n)$.
We now give the definition of weak solutions to (1.1).
Definition 1.1. weak solution
If $u \in {\mathcal{A}}\left( \Omega \right)$ satisfies

for every $\varphi \in {\mathcal{A}}\left( \Omega \right)$ with φ = 0 a.e. in
${\mathbb{H}^n}\backslash \Omega $, then we call u a weak solution to (1.1).
Note that $u\in {\mathcal{A}}( \Omega)$ implies
$u\in{HW}^{s,p}\left( \Omega \right)$, i.e.,
${\mathcal{A}}\left( \Omega \right) \subset {HW}^{s,p}\left( \Omega \right)$. Hence in this work, we only consider the case
$sp \le Q$. Otherwise, the complementary scenario sp > Q ensures the local boundedness and Hölder continuity directly because of the fractional Morrey embedding in the Heisenberg group [Reference Adimurthi and Mallick1].
Our main results are stated as follows. The first one is the local boundedness of weak solutions.
Theorem 1.2 Let the conditions (1.4) and (1.5) be in force. If

then every weak solution $u\in {\mathcal{A}}(\Omega ) \cap L_{sp}^{q - 1}\left({{\mathbb{H}^n}}\right)$ to (1.1) is locally bounded in Ω.
The second one is about the Hölder regularity of weak solutions to (1.1) via supposing $a(\cdot,\cdot)$ is Hölder continuous and the distance between q and p is small. For simplicity, we denote

as the set of basic parameters intervening in the problem.
Theorem 1.3 Let the conditions (1.4)–(1.6) with

be in force. If weak solution $u\in {\mathcal{A}}(\Omega ) \cap L_{sp}^{q - 1}\left({{\mathbb{H}^n}}\right)$ to (1.1) has local boundedness in Ω, then it is locally Hölder continuous as well, that is, for any subset
$\Omega'\subset\subset\Omega$, u belongs to
$C^{0,\beta}_{\rm loc}(\Omega')$ with some
$\beta\in\left(0,\frac{sp}{q-1}\right)$ depending on
$\mathrm{\mathbf{data}}$ and
$\|u\|_{L^\infty(\Omega')}$.
Putting these two theorems above, Hölder continuity is immediately obtained without local boundedness assumption under the intersecting conditions.
Remark 1.4. For the case s > t, local boundedness can be obtained under (1.5), (1.8) by checking the proof of theorem 1.2. Meanwhile, following the proof of theorem 1.3 and making a few slight modifications, we can deduce, under the same preconditions of theorem 1.3, that weak solutions are also of the class $C^{0,\beta}_{\rm loc}(\Omega')$ with some
$\beta\in\left(0,\frac{\min\{sp,tq\}}{q-1}\right)$.
This paper is organized as follows. In $\S$ 2, we introduce the Heisenberg group and function spaces, and then deduce some needful Sobolev embedding theorems. Section 3 is dedicated to proving local boundedness of weak solutions by the Caccioppoli-type estimate. At last, we shall show that the locally bounded weak solutions to (1.1) are Hölder continuous via establishing Logarithmic-type inequality in
$\S$ 4.
2. Functional setting
In this section, we introduce the Heisenberg group ${{\mathbb{H}}^n}$ and some function spaces, and establish several important Sobolev embedding results. The Euclidean space
${{\mathbb{R}}^{2n + 1}}\;(n \ge 1)$ with the group multiplication

where $\xi = \left( {{x_1},{x_2}, \cdots ,{x_{2n}},\tau} \right),$
$\eta = \left( {{y_1},{y_2}, \cdots ,{y_{2n}},\tau'} \right) \in {{\mathbb{R}}^{2n+1}},$ leads to the Heisenberg group
${{\mathbb{H}}^n}$. The left invariant vector field on
${{\mathbb{H}}^n}$ is of the form

and a non-trivial commutator is

We call that ${X_1},{X_2}, \cdots ,{X_{2n}}$ are the horizontal vector fields on
${{\mathbb{H}}^n}$ and T the vertical vector field. Denote the horizontal gradient of a smooth function u on
${{\mathbb{H}}^n}$ by

The Haar measure in ${{\mathbb{H}}^n}$ is equivalent to the Lebesgue measure in
${{\mathbb{R}}^{2n+1}}$. We denote the Lebesgue measure of a measurable set
$E \subset {{\mathbb{H}}^n}$ by
$\left| E \right|$. For
$\xi = \left( {{x_1},{x_2}, \cdots ,{x_{2n}},\tau} \right),$ we define its module as

The Carnot-Carathéodary metric between two points ξ and η in ${{\mathbb{H}}^n}$ is the shortest length of the horizontal curve joining them, denoted by
$d(\xi,\eta)$. The C-C metric is equivalent to the Korànyi metric, i.e.,
$d\left( {\xi,\eta} \right) \sim {\| {{\xi^{- 1}}\circ \eta} \|_{{{\mathbb{H}}^n}}}$. The ball

is defined by the C-C metric d. When not important or clear from the context, we will omit the center as follows: $B_r:=B_r( \xi_0)$.
Let $1 \le p \lt \infty ,\;s \in \left( {0,1} \right)$, and
$v:{{\mathbb{H}}^n} \to {\mathbb{R}}$ be a measurable function. The Gagliardo semi-norm of v is defined as

and the fractional Sobolev spaces $H{W^{s,p}}\left( {{{\mathbb{H}}^n}} \right)$ on the Heisenberg group are defined as

endowed with the natural fractional norm

For any open set $\Omega \subset {{\mathbb{H}}^n}$, we can define similarly fractional Sobolev spaces
$H{W^{s,p}}\left( \Omega \right)$ and fractional norm
${\| v \|_{H{W^{s,p}}\left( \Omega \right)}}$. The space
$HW_0^{s,p}\left( \Omega \right)$ is the closure of
$C_0^\infty \left( \Omega \right)$ in
$H{W^{s,p}}\left( \Omega \right)$. Throughout this paper, we denote a generic positive constant as c or C. If necessary, relevant dependencies on parameters will be illustrated by parentheses, i.e.,
$c=c(n,p)$ means that c depends on
$n,p$. Now we recall the fractional Poincaré type inequality and Sobolev embedding in the Heisenberg group
$\mathbb{H}^n$; see [Reference Piccinini34, proposition 2.7] and [Reference Kassymov and Surgan28, theorem 2.5], respectively.
Lemma 2.1. Poincaré type inequality
Let $ p\ge1,\;s \in \left( {0,1} \right)$ and
$v \in HW^{s,p}( B_r)$. Then we have

where $c=c(n,p) \gt 0$,
${\left( v \right)_r} = -\!\!\!\!\!\!\int_{{B_r}} {vd\xi } $.
Lemma 2.2. Let $1 \lt p \lt \infty ,\;s \in \left( {0,1} \right)$ such that sp < Q. Let also
$v:{{\mathbb{H}}^n} \to {\mathbb{R}}$ be a measurable compactly supported function. Then there is a positive constant
$c = c\left( {n,p,s} \right)$ such that

with ${p_s^*} = \frac{{Qp}}{{Q - sp}}$ being a critical Sobolev exponent.
Now we also give the following result, a truncation lemma near $\partial\Omega$.
Lemma 2.3. Let $ p \ge 1 ,\;s \in \left( {0,1} \right)$ and
$v \in HW^{s,p}\left( {{B_r}} \right)$. If
$\varphi \in {C^{0,1}}\left( {{B_r}} \right) \cap {L^\infty }\left( {{B_r}} \right)$, then it holds that
$\varphi v \in HW^{s,p}\left( {{B_r}} \right)$ and
${\| {\varphi v} \|_{H{W^{s,p}}\left( {{B_r}} \right)}} \le c{\| v \|_{H{W^{s,p}}\left( {{B_r}} \right)}}$ with c > 0 depending on
$n,p,s,r\;\hbox{and}\;\varphi.$
The proof of this lemma is very similar to that of [Reference Di Nezza, Palatucci and Valdinoci15, lemma 5.3], so we omit it here. Based on lemmas 2.1–2.3, we could conclude a Sobolev–Poincaré inequality on balls in the Heisenberg group, which plays a crucial role in proving regularity of solutions.
Proposition 2.4. Sobolev–Poincaré type inequality
Let $1 \lt p \lt \infty ,\;s \in \left( {0,1} \right)$ fulfil sp < Q. Suppose that
$v \in H{W^{s,p}}\left( {{B_R(\xi_0)}} \right)$ and
${B_r(\xi_0)} \subset {B_R(\xi_0)}\;(0 \lt r \lt R)$ are concentric balls. Then there exists a positive constant
$c=c(n,p,s)$ such that

where

Proof. Take $\varphi \left( \xi \right) \in C_0^\infty \left( {{B_R}\left( {{\xi _0}} \right)} \right)$ as a cut-off function such that
$0 \le \varphi \le 1,\;\varphi \equiv 1$ in
${{B_r}\left( {{\xi _0}} \right)}$,
${\rm supp}\, \varphi \subset B_\frac{R+r}{2}( \xi _0)$ and
$\left| {{\nabla _H}\varphi } \right| \le \frac{c}{{R - r}}$ in
${{B_R}\left( {{\xi _0}} \right)}$. Then
$(v-(v)_r)\varphi \in HW_0^{s,p}(B_R)$ and further
$(v-(v)_r)\varphi\in HW_0^{s,p}( {\mathbb{H}^n})$ by zero extension. We split
${\mathbb{H}^n} \times {\mathbb{H}^n}$ into

By virtue of lemma 2.2 and the definition of φ, we get

Note that

We first evaluate J 11 as

where in the last line we have utilized lemma 2.1. On the other hand,

Thus

Moreover, for $\xi \in {{\mathbb{H}^n}\backslash {B_R}}$,
$\eta \in {B_{\frac{{R + r}}{2}}}$, owing to the triangle inequality [Reference Cygan11] there holds that

From this, it follows that

the procedure of which is analogous to J 1. Eventually, we obtain

which implies the statement.
If we let $R=2r$ in the preceding Sobolev–Poincaré inequality, then we can get the very simple version below.
Corollary 2.5. Let $1 \lt p \lt \infty,s \in (0,1)$ fulfil sp < Q. Suppose that
$v\in H{W^{s,p}}(B_{2r})$ and
$B_r \subset B_{2r}$ are concentric balls. Then there exists a positive constant
$c(n,p,s)$ such that

The following result shows an embedding relation between the fractional Sobolev spaces $HW^{t,q}(\Omega)$ and
$HW^{s,p}(\Omega)$.
Lemma 2.6. Let $1 \lt p \le q$ and
$0 \lt s \lt t \lt 1$. Let also Ω be a bounded measurable subset of
$\mathbb{H}^n$. Then there holds that, for each
$v \in HW^{t,q}(\Omega)$,

where c > 0 depends upon $n,p,q,s,t$.
Proof. For p < q, we first utilize the Hölder inequality to get

On the other hand,

with $d: =\mathrm{diam}\left( \Omega \right)$. The combination of preceding inequalities implies the desired display.
If q = p, noting ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \le \mathrm{diam}\left( \Omega \right)$ for
$\xi,\eta \in \Omega$ and s < t, we can readily obtain

Now, we complete the proof.
The forthcoming two lemmas are the consequences of these results above, which will be exploited in the proof of boundedness and Hölder continuity for solutions.
Lemma 2.7. Assume that $s,t\in (0,1)$,
$1 \lt p\le q$ and (1.8) hold. Then for every
$f \in H{W^{s,p}}\left( {{B_r}} \right)$ we infer that

where ${\rm{supp}}\,f: = \{{B_r}:f \ne 0\} $, and c > 0 depends only upon
$n,p,q,s,t$, and
${a_0}$ is any positive constant.
Proof. By the Hölder inequality and proposition 2.4, we obtain

where we used the inequality below,

On the other hand, via the Hölder inequality and proposition 2.4 again,

where we can see that

We finally observe the plain relation that

In summary, we combine all the previous inequalities to arrive at the desired display.
Now denote

Lemma 2.8. Let $s,t \in (0,1)$,
$1 \lt p\le q$ and
$a(\cdot,\cdot)$ satisfy (1.6) and (1.9). Assume
$f \in H{W^{t,q}}\left( {{B_{\bar R}}} \right) \cap {L^\infty }\left( {{B_{\bar R}}} \right)$ with
${\bar R} \le 1$. Then for
$\gamma : = \min \left\{{\frac{{p_s^*}}{p},\frac{{q_t^*}}{q}} \right\} \gt 1$, we have

where ${B_r} \subset {B_R} \subseteq {B_{\bar R}}$ are concentric balls with
$\frac{1}{2}\bar R \le r \lt R \le \bar R$, and c > 0 depends only on
$n,p,q,s,t$ and
${\left[ a \right]_\alpha }$. Here
${\widetilde D}_1(R,r)$ is the corresponding
${D_1}\left( {R,r} \right)$ defined in proposition 2.4 with sp replaced by tq.
Proof. In view of Hölder continuity of a, we have

Then we by employing $tq \le sp+\alpha$,
$r \le 1$ have

Thus

Observe that

Moreover, it follows from proposition 2.4 that

and

Merging the last four inequalities leads to

We now finish the proof.
3. Local boundedness
This section is devoted to showing the interior boundedness of weak solutions to equation (1.1) by means of the key ingredient, a Caccioppoli-type inequality in the nonlocal framework. The forthcoming lemma indicates the multiplication of each function in $\mathcal{A}(\Omega)$ and a cut-off function also belongs to
$\mathcal{A}(\Omega)$.
Lemma 3.1. Let $s,t,p$ and q satisfy (1.4) and
$\varphi \in HW_0^{1,\infty }\left( {{B_r}} \right), v \in {\mathcal A}(\Omega)$. If one of the following two conditions holds:
(i) The inequality (1.8) holds and
$v \in {L^p}\left( {{B_{2r}}} \right)$ satisfies
$\rho \left( {v ;{B_{2r}}} \right) \lt \infty$;
(ii)
$v \in {L^q}\left( {{B_{2r}}} \right)$ satisfies
$\rho \left( {v ;{B_{2r}}} \right) \lt \infty $,
then $\rho \left( {v \varphi ;\mathbb{H}^n} \right) \lt \infty $. In particular,
$v \varphi \in {\mathcal A}(\Omega)$ whenever
$ {B_{2r}}\subset\Omega$.
Proof. By $v \in {\mathcal A}(\Omega)$, proposition 2.4 and (1.8), we get
$v \in {L^q}\left( {{B_{3r/2}}} \right)$ in (i). Thus, we just consider condition (ii). By the definition of
$\rho \left( {v \varphi ;\mathbb{H}^n} \right)$, we have

Owing to $\varphi \in HW_0^{1,\infty }\left( {{B_r}} \right)$, we find

The term I 2 is estimated as

Thus, it follows $\rho \left( {v \varphi ;{\mathbb{H}^n}} \right) \lt \infty$ by combining (3.2), (3.3) with (3.1).
Next, we prove a nonlocal Caccioppoli-type inequality. Define

The numerical inequality below, to be exploited frequently, is from [Reference Di Castro, Kuusi and Palatucci14, lemma 3.1].
Lemma 3.2. Let $p \ge 1$ and
$a,b \ge 0$. Then we have

and

for any $\varepsilon \in \left( {0,1} \right)$ and some
$c=c(p) \gt 0$.
Lemma 3.3. Caccioppoli-type inequality
Let ${B_{2r}}\left( {{\xi _0}} \right) \subset \subset \Omega $,
$1 \lt p\le q$, (1.5) and (1.8) hold. Assume
$u\in {\mathcal{A}}(\Omega )$ is a weak solution to (1.1). Then for any
$\phi \in C_0^\infty \left( {{B_r}} \right)$ with
$0\le \phi \le 1$, we have

for some $c:=c(n,s,t,p,q) \gt 0$, where
${w_\pm }:=(u-k)_ \pm$ with
$k \ge 0$.
Proof. We just consider the estimate for $w_+$, since the estimate for
$w_-$ can be proved similarly. By lemma 3.1, it follows that
${w_+}{\phi ^q} \in {\mathcal{A}}(\Omega )$ from
$u \in {\mathcal{A}}(\Omega )$ and
$\phi \in C_0^\infty \left( {{B_r}} \right) \subset HW_0^{1,\infty} \left( {{B_r}} \right)$, so we can take the testing function
$\varphi={w_+}{\phi ^q}$ in (1.7). Then we have

We first estimate J 1. Since J 1 is symmetry for ξ and η, we may suppose without loss of generality that ${u\left( \xi \right) \ge u\left( \eta \right)}$. Then for
$l \in \{p,q\}$, it yields

Moreover,

which implies

Since

from lemma 3.2, we use Young’s inequality, $0\le \phi\le 1$ and
${\frac{{q - 1}}{q}} \gt 0$ to deduce that

Then, by choosing ɛ small enough, we have

Thus, we get

Now we estimate J 2. Note that

In fact, when $u\left( \xi \right) \ge u\left( \eta \right)$, it easy to see that the inequality (3.8) holds. When
$u\left( \xi \right) \lt u\left( \eta \right)$ and
$u\left( \xi \right)\le k$,
${w_+ }\left( \xi \right)=0$, the inequality (3.8) also holds. When
$k \lt u\left( \xi \right) \lt u\left( \eta \right)$,

Thus, we apply (3.8) and (3.4) to get

The following standard iteration lemma can be found in [Reference Giusti25, lemma 7.1].
Lemma 3.4. Let $\left\{{{y_i}} \right\}_{i = 0}^\infty $ be a sequence of nonnegative numbers satisfying

for some constants $b_1,\;\beta \gt 0$ and
$b_2 \gt 1$. If

then ${y_i} \to 0$ as
$i \to \infty$.
We end this section by providing the proof of boundedness. Lemmas 2.7 and 3.3 play the vital roles in the process.
Proof of theorem 1.2. For convenience, denote

Let ${B_{r}} \equiv {B_{r}}\left( {{\xi _0}} \right) \subset \subset \Omega $ be a fixed ball with
$r \le 1$. For
$i=0,1,2,\cdots$ and
$k_0 \gt 0$, we write

and

In addition, we denote

Then via ${\left( {u\left( \xi \right) - k_i} \right)_ + }\le {\left( {u\left( \xi \right) - k_{i-1}} \right)_ + }$,

Moreover, for $\xi \in {A^ + }\left( {{k_i},{r_i}} \right)$, we have

Thus, it deduces

and

We use lemma 2.7 with $f:= {\left( {u - k} \right)_ + }, a_0:=\|a\|_{L^\infty}$ and (3.11) to get

When we apply lemma 3.3, we choose a cut-off function $\phi \in C_0^\infty \left( {{B_{\frac{{\sigma _i + {r_{i - 1}}}}{2}}}} \right)$ satisfying
$0 \le \phi\le 1,\;\phi\equiv 1 $ in
$B_{\sigma _i}$ and
$\left| {{\nabla _H}\phi } \right| \le \frac{c}{{r_{i-1}-\sigma _i }}=\frac{c}{r}2^i$. Then we have that, from (3.12),

where we used the fact that

and

for ${\xi \in {\rm{supp}}\;\phi }$ and
$\eta \in {{\mathbb{H}^n}\backslash {B_{r_{i - 1}} }}$. Noting that
${D_1}({\sigma _i},{r_i}) \le c{2^{i\left( {Q + p} \right)}}$, it follows from (3.13) that

Since ${H_0}\left( u \right) \in {L^1}\left( \Omega \right)$ from the assumption (1.8), we get that

First, we consider $k_0 \gt 1$ so large that

Then, we have from (3.14) that

where

Finally, we can choose k 0 so large that

holds. Then lemma 3.4 implies

which means that $u \le 2{k_0}$ a.e. in
${B_{\frac{r}{2}}}$.
Applying the same argument to −u, we consequently obtain $u \in {L^\infty }( {{B_{\frac{r}{2}}}} )$.
4. Hölder continuity
We are going to demonstrate the Hölder regularity of weak solutions to equation (1.1) in the last section. First, the second important tool, logarithmic estimate, is established as follows. Throughout this part, we fix any subdomain $\Omega ' \subset \subset \Omega $.
Lemma 4.1. Logarithmic inequality
Let $s,t,p,q$ satisfy (1.4) and
$a(\cdot, \cdot)$ fulfil (1.5), (1.6) with (1.9). Let also
$u\in \mathcal{A}(\Omega)$ be a weak solution of (1.1) such that
$u\in L^{\infty}(\Omega')$ and
$u \ge 0$ in
$B_R: = {B_R}\left( {{\xi _0}} \right) \subset \Omega '$ with
$R\le 1$. Then for any
$0 \lt r\le \frac{R}{2}$ and d > 0,

holds true. Here $K:=1+d^{q-p}+\| u \|_{{L^\infty }\left( {\Omega '} \right)}^{q - p}$ and the constant
$c\ge1$ depends on
$\mathrm{\textbf{data}}$.
Proof. Let us give some notations as below,

and

with $a_\rho ^ + : = \mathop {\sup }\limits_{{B_\rho } \times {B_\rho }} a\left( {\cdot , \cdot } \right)$ and
$\tau \ge 0$.
Consider a cut-off function $\phi\in C_0^\infty \left( {{B_{\frac{{3r}}{2}}}\left( {{\xi _0}} \right)} \right)$ satisfying

Taking the test function $\varphi \left( \xi \right): = \frac{{\phi^q\left( \xi \right)}}{{{g_{2r}}\left( {u\left( \xi \right) + d} \right)}}$, we have from the weak formulation that

with $\bar u: = u + d$.
In what follows, we deal with I 1 in the case $\bar u\left( \xi \right) \ge \bar u\left( \eta \right)$ that is divided into two subcases:

and

If (4.2) occurs, we first observe that

where the first inequality holds naturally when $\phi(\xi) \le \phi(\eta)$. Here, we have used (4.2) and

the details of which can be found in [Reference Byun, Ok and Song4]. Then, combining (4.4) and Young’s inequality yields

where ɛ was chosen as $\frac{{p - 1}}{{{2^{q + 1}}}}$,
$\frac{{\left( {q - 1} \right)p}}{{p - 1}} \gt q$ and c > 0 is independent of a. We proceed to evaluate
${{G_{2r}}\left( {\bar u\left( \eta \right)} \right)}$. For
$\xi,\eta \in B_{2r}$, recalling the Hölder continuity of a, we get

Thus this implies by the facts that $r\le 1$ and
$tq \le sp+\alpha$ that

Next, we will obtain an estimate on $\log \bar u$. It is easy to find

so, by the monotonicity of the function $f(\tau)=(\tau^p+a(\xi,\eta)\tau^q\|\eta ^{-1}\circ \xi \|_{\mathbb{H}^n}^{-(t-s)q})/\tau$ with
$\tau\ge0$,

where we need to note ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \le 4r$. It follows from (4.5)–(4.7) that

Second, we in the case (4.3) tackle the integral I 1. Applying lemma 3.2 and the relation $\bar u\left( \xi \right) \ge 2\bar u\left( \eta \right)$, we could derive

with $\varepsilon = \frac{{{2^{p - 1}} - 1}}{{{2^p}}}$. Thereby, it holds that

Here $F\left( {\xi ,\eta } \right)$ is the same as that in (4.5) and the estimate for
${{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}$ is similar to (4.6). Moreover, via
$\bar u\left( \xi \right) \ge 2\bar u\left( \eta \right)\ge 0$ in
$B_{2r}$,

and further

Now we obtain an estimate on $\log \frac{{\bar u\left( \xi \right)}}{{\bar u\left( \eta \right)}}$ under (4.3). Notice
$\bar u(\xi)\le 2(\bar u( \xi) - \bar u( \eta))$. we get

where the fact ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \le 4r$ was utilized. Noting
$q\ge p$ and
${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \le 4r$ again,

At this moment, for $\bar u\left( \xi \right) \ge \bar u\left( \eta \right)$, the integral I 1 is evaluated as

where

Furthermore, if $\bar u\left( \xi \right) \lt \bar u\left( \eta \right)$, the same estimate still holds true through exchanging the roles of ξ and η.
For the second contribution I 2 in (4.1), we first observe that if $\eta \in {B_R}$, then
${( {u\left( \xi \right) - u\left( \eta \right)} )_ + }$
$\le u\left( \xi \right) + d$ by
$u\left( \eta \right) \ge 0$, and that if
$\eta \in {\mathbb{H}^n}\backslash {B_R}$, then
${\left( {u\left( \xi \right) - u\left( \eta \right)} \right)_ + } \le u\left( \xi \right) + u_-\left( \eta \right) \le \bar u\left( \xi \right) + u_-\left( \eta \right)$. From this and
${\rm supp}\, \phi\subset {B_{\frac{{3r}}{2}}}$, we can evaluate I 2 as

We now intend to control precisely the term $\frac{{h\left( {\xi ,\eta ,\bar u\left( \xi \right)} \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}}$ by some constants. In view of the condition (1.6), there holds that, for
$\xi \in B_{2r}$ and
$\eta \in \mathbb{H}^n$,

This indicates

by virtue of ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \gt \frac{r}{2}$. For
$\xi \in {B_{\frac{{3r}}{2}}}$ and
$\eta \in \mathbb{H}^n\backslash {B_{2r}}$, via the triangle inequality,

Thus by [Reference Manfredini, Palatucci, Piccinini and Polidoro31, Lemma 2.6],

Let us proceed to examine I 22. With the aid of (4.10), (4.11) and $u(\xi)\ge 0$ in
${{B_{\frac{{3r}}{2}}}}$,

where we notice $\eta \in {\mathbb{H}^n}\backslash {B_R} \subset {\mathbb{H}^n}\backslash {B_{2r}}$.
Merging (4.8), (4.9), (4.12), (4.13) with (4.1) arrives eventually at the desired estimate with the positive constant c depending upon $n,p,q,s,t,\alpha,[a]_{\alpha} $ and
$\| a \|_{L^\infty }$.
Corollary 4.2. Let the assumptions of lemma 4.1 be in force. Define

with $\tau,d \gt 0$ and b > 1. Then for the weak solution u of (1.1) it holds that

where c > 1 depends on $\mathrm{\textbf{data}}$, and K is defined as in lemma 4.1.
Proof. Notice that, since w is a truncation of $\log (u+d)$,

Then the desired result is a plain consequence of lemma 4.1.
In the end, we will focus on establishing Hölder regularity of weak solutions. For this aim, it is sufficient to show an oscillation improvement result, theorem 4.3. Before proceeding, let us introduce some notations. For $j\in\mathbb{N}\cup\{0\}$, set

where we fix any ball $B_{2r}(\xi_0)\subset\Omega'\subset\subset\Omega$. Furthermore, define

and

Let us point out that σ and β are to be determined later.
Now we are in a position to prove the following iteration lemma, which suggests $u\in C^{0,\beta}(B_r)$.
Theorem 4.3 Let $u\in \mathcal{A}(\Omega)\cap L^{q-1}_{sp}(\mathbb{H}^n)$ be a weak solution to (1.1). Under the conditions (1.4), (1.5) and (1.6) with
$tq\le sp+\alpha$, there holds that

where these notations are fixed as above.
Proof. Argue by induction. The conclusion is obvious for j = 0 and then assume it holds true for $i\le j$. Now we show this claim for j + 1. Let us notice the simple fact that either

or

Define

Obviously, $u_j\ge0$ in Bj and

Moreover, uj is a weak solution to (1.1) such that

Now we set an auxiliary function

Applying corollary 4.2 derives

with K defined as in lemma 4.1. We evaluate the second integral at the right-hand side. By means of (4.17) and the definition of $\omega(r_0)$,

where we used the fact that $\beta \lt \frac{sp}{q-1}\left(\le \frac{tq}{q-1}\right)$. Analogously,

with $\beta \lt \frac{sp}{q-1}\left(\le \frac{sp}{p-1}\right)$, where
$N:=1+\|u\|^{q-p}_{L^\infty(\Omega')}$ and the derivation of
$\|u\|^{q-p}_{L^\infty(\Omega')}$ is from the term
$|u_j|^{q-1}$, and C > 0 depends on
$n,p,s$ and the difference of
$\frac{sp}{p-1}$ and β. Combining (4.19), (4.20) with (4.18) and remembering
$\frac{r_{j+1}}{r_j}=\sigma$, we get

where C depends on $n,p,q,s,t$ and the difference of β and
$\frac{tq}{q-1}$, and
$\frac{sp}{p-1}$.
In what follows, picking

and recalling $\omega(r_j)=\sigma^{j\beta}\omega(r_0)$, we find

where C depends on $n,p,q,s,t,\alpha,[a]_\alpha,\|a\|_{L^\infty}$ and the difference of β and
$\frac{tq}{q-1}$, and
$\frac{sp}{p-1}$. Here we need to utilize the definition of K as in lemma 4.1, and
$\omega(r_j)\le 2\|u\|_{L^\infty(\Omega')}$. From the last inequality,

We refer to [Reference Di Castro, Kuusi and Palatucci14, page 1296] for the details. By taking

with $\varepsilon:=\sigma^{\frac{sp}{q-1}-\beta}$, it holds that

for the constant $C_{\rm log} \gt 0$ depending on
$n,p,q,s,t,\alpha,[a]_\alpha,\|a\|_{L^\infty}$ and β.
At this moment, we are going to perform a suitable iteration. For each $i=0,1,\cdots$, let

and the corresponding balls

Then take the cut-off functions $\psi_i\in C^\infty_0(\tilde{B}^i)$ such that

Besides, set

and

Observe the apparent facts that

and denote

With the help of Caccioppoli inequality (lemma 3.3), we derive

Via the definition of wi and ψi, J 1 is evaluated as

and moreover, we have

As for the nonlocal integral in J 2, we first note that if $\eta\in\tilde{B}^i$ and
$\xi\in\mathbb{H}^n\setminus B^i$, then

Furthermore, $w_i\le k_i\le2\varepsilon\omega(r_j)$ in Bj (by
$u_j\ge0$ in Bj), and
$w_i\le k_i+|u|$ in
$\mathbb{H}^n\setminus B_j$. In a similar way to treat I 2 in the proof of lemma 4.1, by applying (4.19), (4.20), the definition of ɛ and
$B_{j+1}\subset B^i$ we derive

Therefore,

On the other hand, making use of lemma 2.8 with $u:=w_i$ yields that

Thanks to the definitions of $D_1,\widetilde{D}_1$ and
$\hat{\rho}_i,\rho_{i+1}$, we from
$\hat{\rho}_i\approx\rho_{i+1}\approx r_{j+1}$ and
$\hat{\rho}_i-\rho_{i+1}=2^{-i-3}r_{j+1}$ calculate

It is easy to obtain

It follows from (4.22)–(4.26) that

and further

where $\gamma=\min\left\{\frac{p^*_s}{p},\frac{q^*_t}{q}\right\} \gt 1$ and C depends on
$\mathrm{\textbf{data}}$ and β.
Now if A 0 fulfils

then by lemma 3.4 we deduce $A_i\rightarrow0$ as
$i\rightarrow\infty$. This means

which together with (4.17) leads to

Finally, choosing $\beta\in\left(0,\frac{sp}{q-1}\right)$ small enough such that

then $\mathrm{osc}_{B_{j+1}}u\leq\omega(r_{j+1})$, and β depends on
$\mathrm{\textbf{data}}$ and
$\|u\|_{L^\infty(\Omega')}$. Indeed, due to (4.21), it yields that

by picking $\sigma\le \mathrm{exp}\left(-\frac{C_{\rm log}N^3}{\mu}\right)$. Then, we select
$\sigma=\min\left\{\frac{1}{4},\mathrm{exp}\left(-\frac{C_{\rm log}N^3}{\mu}\right)\right\}$ to ensure the condition (4.27) does hold true. Now we finish the proof.
Acknowledgements
The authors wish to thank the anonymous reviewer for valuable comments and suggestions to improve the manuscript. This work was supported by the National Natural Science Foundation of China (No. 12071098), the Postdoctoral Science Foundation of Heilongjiang Province (No. LBH-Z22177), the National Postdoctoral Program for Innovative Talents of China (No. BX20220381), the Fundamental Research Funds for the Central Universities (No. 2022FRFK060022) and the Natural Science Basic Research Program of Shaanxi (No. 2024JC-YBQN-0054).
Declarations
Conflict of interest
The authors declare that there is no conflict of interest. We also declare that this manuscript has no associated data.
Data availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.