Amiran Gogatishvili

Boundedness of the fractional maximal operator in local Morrey-type spaces

The problem of boundedness of the fractional maximal operator M α, 0 ≤ α < n, in general local Morrey-type spaces is reduced to the problem of boundedness of the supremal operator in weighted L p -spaces on the cone of non-negative non-decreasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.

Characterizations of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces

Abstract. On a metric measure space satisfying the doubling property, we establish several optimal characterizations of Besov and Triebel–Lizorkin spaces, including a pointwise characterization. Moreover, we discuss their (non)triviality under a Poincaré inequality.

Characterisation of Embeddings in Lorentz Spaces

Characterization of embeddings in Lorentz spaces using a method of discretization and anti-discretization

Embeddings and Duality Theorems for Weak Classical Lorentz Spaces

We characterize the weight functions u, v, w on (0, ∞) such that �Z 1 0 f � (t) q w(t) dt � 1/q ≤ C sup t2(0,1) f �� u (t)v(t),

Some New Iterated Hardy-Type Inequalities

We characterize the validity of the Hardy-type inequality , where , , , , , and are weight functions on . Some fairly new discretizing and antidiscretizing techniques of independent interest are used.

SOLUTION OF TWO-WEIGHT PROBLEMS FOR INTEGRAL TRANSFORMS WITH POSITIVE KERNELS

Criteria for weak and strong two-weighted inequalities are obtained for integral transforms with positive kernels.

CRITERIA OF GENERAL WEAK TYPE INEQUALITIES FOR INTEGRAL TRANSFORMS WITH POSITIVE KERNELS

AbstractNecessary and sufficient conditions are derived in order that an inequality of the form

Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness

\begin{gathered} \varphi (\lambda )\theta (\beta \{ (x,t) \in X x (0,\infty ):\kappa (fdv)(x,t) > \lambda \} ) \leqslant \hfill \\ \leqslant c\int\limits_X {\psi \left( {\frac{{f(x)}}{{\eta (\lambda )}}} \right)\sigma (x)dv(x)} \hfill \\ \end{gathered}