Narrow Results

We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property.

We prove global boundedness and Hölder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical $L^p$-result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.

Let $\lambda \in (0,\infty )$ and ${△}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator on ${ℝ}_{+}:=(0,\infty )$. In this paper, the authors show that $b\in \mathrm{\text{BMO}}({ℝ}_{+},d{m}_{\lambda })$ (or $\mathrm{\text{CMO}}({ℝ}_{+},d{m}_{\lambda })$, respectively) if and only if the Riesz transform commutator $[b,R_{{△}_{\lambda }}]$ is bounded (or compact, respectively) on Morrey spaces $L^{p,\kappa }({ℝ}_{+},d{m}_{\lambda })$, where $d{m}_{\lambda }(x):=x^{2\lambda }dx$, $p\in (1,\infty )$ and $\kappa \in (0,1)$. A weak factorization theorem for functions belonging to the Hardy space $H^1({ℝ}_{+},d{m}_{\lambda })$ in the sense of Coifman–Rochberg–Weiss in Bessel setting, via $R_{{△}_{\lambda }}$ and its adjoint, is also obtained.

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on ${ℝ}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley $g$-functions, Lusin area functions, Littlewood–Paley $g_{\lambda }^{\ast }$-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley $g$-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on ${ℝ}^n$ in the aforementioned function spaces.

In this paper, we study the continuous fractional wavelet transform (CFrWT) in $n$-dimensional Euclidean space ${ℝ}^n$ with scaling parameter $a=(a_1,a_2,\dots ,a_n)\in {ℝ}^n$ such that $a_i\ne 0,\forall i=1(1)n$. We obtain inner product relation and reconstruction formula for the CFrWT depending on two wavelets along with the reproducing kernel function, involving two wavelets, for the image space of CFrWT. We obtain Heisenberg’s uncertainty inequality and Local uncertainty inequality for the CFrWT. Finally, we prove the boundedness of CFrWT on the Morrey space $L_M^{1,\nu }({ℝ}^n)$ and estimate $L_M^{1,\nu }({ℝ}^n)$-distance of the CFrWT of two argument functions with respect to different wavelets.

We study the Navier–Stokes equations of three-dimensional compressible isentropic and two-dimensional heat-conducting flows in a domain Ω with nonnegative density, which may vanish in an open subset (vacuum) of Ω, and with positive density, respectively. We prove some blow-up criteria for the local strong solutions.