Related papers: Weighted maximal $L_{q}(L_{p})$-regularity theory …

In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $$ \partial_t u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ Here, $\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents…

In this study, we investigate the existence, uniqueness, and maximal regularity estimates of solutions to homogeneous initial value problems involving time-measurable pseudo-differential operators within the framework of weighted mixed norm…

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the…

We prove maximal $L^p$-regularity for the stochastic evolution equation \[\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}.\] under the assumption that $A$ is a sectorial…

We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common H\"older assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In…

In \cite{MR447956}, Muckenhoupt and Wheeden formulated a weighted weak $(p,p)$ inequality where the weight for the weak $L^p$ space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood…

We consider an evolution equation involving the fractional powers, of order $s \in (0,1)$, of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order $\gamma \in (1,2]$. Since it has been…

We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.

Assuming $A$ has maximal $L^p$-regularity, this paper investigates perturbations of $A$ by time-dependent operators $B$ that are unbounded and satisfy a critical $L^q$-integrability condition in time. We establish two main results. The…

For $\alpha >1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-\Delta)^{\alpha/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values…

Let $n\ge 2$ be the spatial dimension. The purpose of this note is to obtain some weighted estimates for the fractional maximal operator ${\mathfrak M}{\alpha}$ of order $\alpha$, $0\le\alpha<n$, on the weighted Choquet-Lorentz space…

In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag.…

We develop a maximal regularity approach in temporally weighted $L_p$-spaces for vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions, both of static and of relaxation type. Normal ellipticity and…

In this work, we consider a time-fractional Allen-Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order $\alpha\in(0,1)$. First, the well-posedness and (limited) smoothing…

For $q>2, \gamma > 1$, we prove that maximal regularity of $L^q$ type holds for periodic solutions to $-\Delta u + |Du|^\gamma = f$ in $\mathbb{R}^d$, under the (sharp) assumption $q > d \frac{\gamma-1}\gamma$.

We prove two maximal regularity results in spaces of continuous and H\"older continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^\alpha$. This derivative is intended in the sense…

Taking into account the asymptotic behavior of some Wright functions and the existence of bounds for the Mainardi and the Wright function $W(-x,\frac{\alpha}{2}, 1)$ in $\mathbb{R}^+$ , three different initial-boundary-value problems for…

In this paper we establish weighted $L^{q}$-$L^{p}$-maximal regularity for linear vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in…

This paper studies a maximal $L^q$-regularity property for nonlinear elliptic equations of second order with a zero-th order term and gradient nonlinearities having superlinear and sub-quadratic growth, complemented with Dirichlet boundary…

We study the fractional $(p,q)$-Laplace equation $$ (-\Delta_p)^s u +(-\Delta_q)^t u= 0 $$ for $s,t\in(0,1)$ and $p,q\in(1,\infty)$. We establish H\"older estimates with an explicit exponent. As a consequence, we derive a Liouville-type…