Related papers: On the Schrodinger maximal function in higher dime…
Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schr\"odinger equation are considered. Due to the fact that…
A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie…
Our main result is an estimate for a sharp maximal function, which implies a Keith-Zhong type self-improvement property of Poincar\'e inequalities related to differentiable structures on metric measure spaces. As an application, we give…
We systematically describe and classify 1-dimensional Schr\"odinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe 2 new classes of exactly solvable…
We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is…
We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.
We present new sharp results concerning multipliers and distance estimates in various spaces of harmonic functions in the unit ball of $R^n$.
Various optimal estimates for solutions of the Laplace, Lam\'e and Stokes equations in multidimensional domains, as well as new real-part theorems for analytic functions are obtained.
This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…
We obtain $L^p-$estimates for the full and lacunary maximal functions associated to the twisted bilinear spherical averages given by \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;d\sigma(z_1,z_2),\;t>0,\]…
For p>1 we find the Bellman function of two variables associated with the dyadic maximal operator on Rn.Actually we do that in the more general setting of tree-like maximal operators.We provide a simple and elementary proof,different from…
We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\infty$ and…
This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical,…
We show that the Schr\"odinger operator associated with a physical system over a local field can be approximated in a very strong sense by finite Schr\"odinger operators. Some striking numerical results are included at the end of the…
In this note we describe some results concerning upper and lower bounds for the Jensen functional. We use several known and new results to shed light on the concepts of superterzatic functions.
For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the…
We look for positive solutions to the nonlinear Schrodinger equation with a potential, under the hypothesis of zero mass on the nonlinearity, in a particular situation. Existence and multiplicity results are provided.
New sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper.
H\"older estimates for second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.
We study pointwise convergence of the fractional Schr\"odinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-\Delta)^{\alpha/2}} f| $ can be deduced from those…