The common topic of this thesis is boundedness of integral and supremal operators between function spaces with weights. The results of this work have the form of characterizations of validity of weighted operator inequalities for appropriate cones of functions. The outcome can be divided into three categories according to the particular type of studied operators and function spaces. The first p...

Abstract By the use of weight coefficients, idea introduced parameters and technique real analysis, a more accurate Hilbert-type inequality in whole plane with general homogeneous kernel is given, which an extension Hardy–Hilbert’s inequality. An equivalent form obtained. The statements best possible constant factor related to several parameters, operator expressions few particular cases are co...

Abstract In this work, by the introduction of some parameters, a new half-discrete kernel function in whole plane is defined, which involves both homogeneous and nonhomogeneous cases. By employing techniques real analysis, especially method weight function, Hilbert-type inequality with as well its equivalent Hardy-type inequalities are established. Moreover, it proved that constant factors newl...

We investigate Bergman type operators on the complex unit ball, which are singular integral induced by modified kernel. consider $L^p$-$L^q$ boundedness and compactness of operators. The results can be viewed as Hardy–Littlewood–Sobolev (HLS) theorem in case ball. also give some sharp norm estimates fact gives upper bounds optimal constants HLS inequality Moreover, a trace formula is given.

For $\alpha > 0$ we consider the operator $K_\alpha \colon \ell^2 \to \ell^2$ corresponding to matrix \[\left(\frac{(nm)^{-\frac{1}{2}+\alpha}}{[\max(n,m)]^{2\alpha}}\right)_{n,m=1}^\infty.\] By interpreting $K_\alpha$ as inverse of an unbounded Jacobi matrix, show that absolutely continuous spectrum coincides with $[0, 2/\alpha]$ (multiplicity one), and there is no singular spectrum. There a f...

We show that a number of well known multiplier theorems for Hardy spaces over Vilenkin groups follow immediately from a general condition on the kernel of the multiplier operator. In the compact case, this result shows that the multiplier theorems of Kitada [6], Tateoka [13], Daly-Phillips [2], and Simon [11] are best viewed as providing conditions on the partial sums of the Fourier-Vilenkin se...

Let μ be a nonnegative Radon measure on R which only satisfies the following growth condition that there exists a positive constant C such that μ B x, r ≤ Cr for all x ∈ R, r > 0 and some fixed n ∈ 0, d . In this paper, the authors prove that for suitable indexes ρ and λ, the parametrized g∗ λ function M∗,ρ λ is bounded on L μ for p ∈ 2,∞ with the assumption that the kernel of the operator M∗,ρ...

The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover , the Hardy-Littlewood-Sobolev inequality is established.