We discuss the dyadic John–Nirenberg space that is a generalization of functions of bounded mean oscillation. A John–Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John–Nirenberg space and provide a method to construct nontrivial functions in the dyadic John–Nirenberg space. Moreover, we prove that the John–Nirenberg space is complete. Several open problems are also discussed.

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Let L be a one-to-one operator of type ω in L2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Let p(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H_L^{p(\cdot )} ({\open R}^n)$ associated with L. By means of variable tent spaces, the authors establish the molecular characterization of $H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of $H_L^{p(\cdot )} ({\open R}^n)$ is the bounded mean oscillation (BMO)-type space ${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, where L* denotes the adjoint operator of L. In particular, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H_L^{p(\cdot )} ({\open R}^n)$ and show that the fractional integral L−α for α∈(0, (1/2)] is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to $H_L^{q(\cdot )} ({\open R}^n)$ with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇ L−1/2 is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to the variable Hardy space Hp(·)(ℝn).

In this paper we introduce Hardy-Lorentz spaces with variable exponents associated with dilations in ${{\mathbb{R}}^{n}}$. We establish maximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.

In this note we give a simple proof of the endpoint regularity for the uncentred Hardy–Littlewood maximal function on $\mathbb{R}$. Our proof is based on identities for the local maximum points of the corresponding maximal functions, which are of interest in their own right.

Let $w$ be either in the Muckenhoupt class of ${{A}_{2}}\left( {{\mathbb{R}}^{n}} \right)$ weights or in the class of $QC\left( {{\mathbb{R}}^{n}} \right)$ weights, and let ${{L}_{w}}\,:=\,-{{w}^{-1}}\,\text{div}\left( A\nabla \right)$ be the degenerate elliptic operator on the Euclidean space ${{\mathbb{R}}^{n}}$, $n\,\ge \,2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ associated with ${{L}_{w}}$ for $p\,\in \,(0,\,1]$, and when $p\,\in \,(\frac{n}{n+1},\,1]$ and $w\,\in \,{{A}_{{{q}_{0}}}}\left( {{\mathbb{R}}^{n}} \right)$ with ${{q}_{0}}\,\in \,[1,\,\frac{p(n+1)}{n})$, the authors prove that the associated Riesz transform $\nabla L_{w}^{-1/2}$ is bounded from $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ to the weighted classical Hardy space $H_{w}^{p}\left( {{\mathbb{R}}^{n}} \right)$.

In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the Hardy–Littlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type (1, 1) estimates for these maximal functions. The ${{L}^{p}}$ estimates for $p\,>\,1$ follow fromthe interpolation. Some applications are included.

Extending previous results of the first author, some new estimates are obtained for maximal operators of Schrödinger type with a complex parameter.

We establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.

We establish an approximation theorem for a sequence oflinear elastic strains approaching a compact set in L 1 by thesequence of linear strains of mapping bounded in Sobolev space W 1,p . We apply this result to establish equalities forsemiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left( \nabla _{\alpha}u_\varepsilon\big| \frac{1}{\varepsilon}\nabla _3 u_\varepsilon\right) $ bounded in $L^p (\Omega; \mathbb{R}^9 ), \ 1 < p < +\infty .$ Here it is shown that, up to a subsequence, $u_\varepsilon$ may be decomposed as $w_\varepsilon+ z_\varepsilon,$ where $z_\varepsilon$ carries all the concentration effects, i.e.$\left\{ \left| \left( \nabla _{\alpha }w_\varepsilon| \frac{1}{\varepsilon }\nabla _3 w_\varepsilon\right) \right| ^{p} \right\} $ is equi-integrable, and $w_\varepsilon$ captures the oscillatory behavior, i.e.$z_\varepsilon\to 0$ in measure. In addition, if $\{ u_\varepsilon\} $ is a recovering sequence then $z_\varepsilon= z_\varepsilon(x_\alpha )$ nearby $\partial \Omega.$

We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on ${{L}^{2}}$ if the Cantor set has positive Hausdorff dimension.