Related papers: Sharp estimates for maximal operators associated t…

We study the maximal operator on the variable exponent H\"older spaces in the setting of metric measure spaces. The boundedness is proven for metric measure spaces satisfying an annular decay property. Let us stress that there are no…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=\Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous…

The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue…

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$.…

We construct homotopy formulas for the $\overline\partial$-equation on convex domains of finite type that have optimal Sobolev and H\"older estimates. For a bounded smooth finite type convex domain $\Omega\subset\mathbb C^n$ that has…

In this paper we consider the wave operators $W_{\pm}$ for a Schr\"odinger operator $H$ in ${\bf{R}}^n$ with $n\geq 4$ even and we discuss the $L^p$ boundedness of $W_{\pm}$ assuming a suitable decay at infinity of the potential $V$. The…

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^\alpha$ of (complex) order $\alpha$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We…

We study pointwise convergence of the solution to the elastic wave equation to the initial data which lies in the Sobolev spaces. We prove that the solution converges along every lines to the initial data almost everywhere whenever the…

We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse…

We study a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. As the main results, we find conditions for the associated maximal operator and its local variant to be bounded on…

We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave operator, from the restriction type estimates we establish the…

This paper investigates the $L^p$-boundedness of wave operators associated with the nonhomogeneous fourth-order Sch\"odinger operator $H = \Delta^2 - \Delta + V(x)$ on $\mathbb{R}^n$. Assuming the real-valued potential $ V $ exhibits…

We prove that the wave operators for $n \times n$ matrix Schr\"odinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L^p(\mathbb R^+, \mathbb C^n), 1 < p < \infty, $ for slowly decaying…

We obtain the optimal value of the constant K(n,s) in the Sobolev-Nirenberg-Gagliardo inequality $ \|\,u\,\|_{L^{\infty}(\mathbb{R}^{n})} \leq K(n,s) \,\|\, u \,\|_{L^{2}(\mathbb{R}^{n})}^{1 - n/(2s)} \|\, u…

We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights $\varphi_{t,\gamma}(\tau)=|(\tau-t)^\gamma|$, where $\gamma$ is a complex number, over arbitrary Carleson curves. If the…

We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the…

We consider a type of maximal operators associated to moment curves in $\mathbb R^d, d\geq 3.$ We derive $L^p$ mapping properties for these operators. In a special case, the estimate is sharp.

Let $\phi$ be a quasiconformal mapping, and let $T_\phi$ be the composition operator which maps $f$ to $f\circ\phi$. Since $\phi$ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins…

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when…

Let $M$ be the maximal operator associated to a smooth curve in $\mathbb R^3$ which has nonvanishing curvature and torsion. We prove that $M$ is bounded on $L^p$ if and only if $p>3$.