On Hardy q-inequalities

Some q-analysis variants of Hardy type inequalities of the form \int_0^b (x^{\alpha-1} \int_0^x t^{-\alpha} f(t) d_qt)^p d_qx \leq C \int_0^b f^p(t) d_qt with sharp constant C are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.Comment: To appear in Czechoslovak Math. J., 22 page

Weighted multidimensional inequalities for monotone functions

summary:We discuss the characterization of the inequality \biggl(\int_{{\Bbb R}^N_+} f^q u\biggr)^{1/q} \leq C \biggl(\int_{{\Bbb R}^N_+} f^p v \biggr)^{1/p},\quad0<q, p <\infty, for monotone functions $f\geq0$ and nonnegative weights $u$ and $v$ and $N\geq1$. We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions

Some new (Hp – Lp) type inequalities for weighted maximal operators of partial sums of Walsh–Fourier series

In this paper we introduce some new weighted maximal operators of the partial sums of the Walsh–Fourier series. We prove that for some “optimal” weights these new operators indeed are bounded from the martingale Hardy space Hp(G) to the Lebesgue space Lp(G), for 0&#60;p&#60;1. Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results

A new discrete Hardy-type inequality with kernels and monotone functions

Link to publishers version: 10.1186/s13660-015-0843-9A new discrete Hardy-type inequality with kernels and monotone functions is proved for the case 1<q<p<∞1<q<p<∞. This result is discussed in a general framework and some applications related to Hölder’s summation method are pointed out

Weighted hardy operators in complementary morrey spaces

We study the weighted p -> q-boundedness of the multidimensional weighted Hardy-type operators H-omega(alpha) and H-omega(alpha) with radial type weight omega = omega(vertical bar x vertical bar), in the generalized complementary Morrey spaces (C) L-(0)(p,psi) (R-n) defined by an almost increasing function psi = psi(r). We prove a theorem which provides conditions, in terms of some integral inequalities imposed on psi and omega, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the function psi and the weight omega are power functions. We also prove that the spaces (C) L-(0)(p,psi) (Omega) over bounded domains Omega are embedded between weighted Lebesgue space L-p with the weight psi and such a space with the weight psi, perturbed by a logarithmic factor. Both the embeddings are sharp.Lulea University of Technolog