We apply capacities to explore the space–time fractional dissipative equation: (0.1)

$$ \begin{align} \left\{\begin{aligned} &\partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x)+f(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+},\\ &u(0,x)=\varphi(x),\ x\in\mathbb R^{n}, \end{aligned}\right. \end{align} $$

where $\alpha>n$ and $\beta \in (0,1)$. In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term $R_{\alpha ,\beta }(\varphi )$ and inhomogeneous term $G_{\alpha ,\beta }(g)$, respectively. Second, we obtain some space–time estimates for $G_{\alpha ,\beta }(g).$ Based on these estimates, we prove that the continuity of $R_{\alpha ,\beta }(\varphi )(t,x)$ and the Hölder continuity of $G_{\alpha ,\beta }(g)(t,x)$ on $\mathbb {R}^{1+n}_+,$ which implies a Moser–Trudinger-type estimate for $G_{\alpha ,\beta }.$ Then, for a newly introduced $L^{q}_{t}L^p_{x}$-capacity related to the space–time fractional dissipative operator $\partial ^{\beta }_{t}+(-\Delta )^{\alpha /2},$ we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in $\mathbb {R}^{1+n}_+$ by using the Strichartz estimates and the Moser–Trudinger-type estimate for $G_{\alpha ,\beta }.$ A strong-type estimate of the $L^{q}_{t}L^p_{x}$-capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the $L^{q}_{t}L^p_{x}$-capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).

Baker and Rumely, Favre, Rivera, and Letelier, and Chambert-Loir proved an important arithmetic equidistribution theorem for points of small height associated to an adelic measure. To broaden the scope in which arithmetic equidistribution may be employed, we generalize the notion of an adelic measure to that of a quasi-adelic measure and show that arithmetic equidistribution holds for quasi-adelic measures as well. We exhibit examples of non-adelic, quasi-adelic measures arising from the dynamics of quadratic rational maps. In fact, we show that the measures that arise in applications of arithmetic equidistribution theorems are typically not adelic. Finally, we motivate our definition of a quasi-adelic measure by relating it to a seemingly different problem in arithmetic dynamics arising from results of Call, Tate, and Silverman in the study of abelian varieties.

We show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

In this paper we characterize the boundedness on the product of Sobolev spaces Hs(𝕋) × Hs(𝕋) on the unit circle 𝕋, of the bilinear form Λb with symbol b ∈ Hs(𝕋) given by

$${{\Lambda}_b} (\varphi, \psi): = \int_{\open T} {\left( {{{( - {\Delta })}^s} + I} \right)(\varphi \psi )(\eta )b(\eta ) {\rm d}\sigma (\eta ).}$$

We prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that the Lp–Lq estimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.

In this note we study the convergence of sequences of Monge–Ampère measures $\{{{(d{{d}^{c}}{{u}_{s}})}^{n}}\}$, where $\{{{u}_{s}}\}$ is a given sequence of plurisubharmonic functions, converging in capacity.

We study negative powers of Laguerre differential operators in ${{\mathbb{R}}^{d}},\,d\,\ge \,1$. For these operators we prove two-weight $[{{L}^{p}}\,-\,{{L}^{q}}$ estimates with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices. we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the convexity principle to cover the continuous range of $\alpha \,\in \,\,{{[-1/2,\infty )}^{d}}$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on ${{\mathbb{R}}^{d}}$ and isomorphic to $\mathbb{Z}_{2}^{d}$. The two weight ${{L}^{p}}\,-\,{{L}^{q}}$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

This paper investigates the tightness property of the capacity induced by the reduction operator with respect to the resolvent of a right Markov process. Tightness is verified in two particular situations: under the ‘weak duality hypothesis’, and if a substitute for ‘the axiom of polarity for the dual theory’ holds. In the second context, the quasi-continuity property for the excessive functions is derived. These are extensions and improvements of results of Lyons and Röckner, Ma and Röckner, Le Jan, and Fitzsimmons, mainly obtained in the context of Dirichlet forms.

The main goal of this paper is to establish new uniform estimates on the size of sublevel sets of plurisubharmonic functions (called plurisubharmonic lemniscates) in terms of Hausdorff–Riesz measures and capacities of certain orders.

We first prove a new uniform version of Skoda's integrability theorem for a given class of plurisubharmonic functions in terms of Borel measures of Hausdorff–Riesz type of certain orders with a precise estimate of the integrability exponent in terms of Lelong numbers of the class and the order of the measures.

Then we present several applications of this result. We first deduce uniform estimates on the size of plurisubharmonic lemniscates associated to functions from some important classes of plurisubharmonic functions in terms of Hausdorff–Riesz measures.

We also derive a new comparison inequality between certain Hausdorff–Riesz capacities and the pluricomplex logarithmic capacity for borelean sets of a fixed bounded domain in $\mathbb{C}^n$ or more generally in an affine algebraic manifold.

Furthermore, using results from classical potential theory, we finally deduce from this comparison inequality new estimates of the size of polynomial lemniscates in terms of Hausdorff contents in the spirit of the famous lemma of H. Cartan.

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

In this paper we study the q-Carleson measures for a space of M-harmonic potentials in the unit ball of Cn, when q < p. We obtain some computable sufficient conditions, and study the relations among them.

For we show that the conditions for all open sets G, C2,q denoting Bessel capacity, are not sufficient to characterize the compact sets X with the property that each function harmonic on and in Lp(X) is the limit in the Lp norm of a sequence of functions which are harmonic on neighbourhoods of X.

We study positive measures on 𝔹n satisfying that . for any ƒ ∈ where is the Hardy-Sobolev space in the unit ball. We obtain several computable sufficient conditions as well as some necessary conditions and establish their sharpness. We study the same problem for Besov-Sobolev spaces and give some applications to multipliers.

Coercive closed forms on L2-spaces are studied whose associated L2-semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.