We study $L^p$ norm convergence of Bochner-Riesz means $S_R^{\delta }f$ associated with certain non-negative differential operators. When the kernel $S_R^m(x,y)$ satisfies a weak estimate for large values of m we prove $L^p$ norm convergence of $S_R^{\delta }f$ for δ > n|1/p-1/2|, 1 < p < ∞, where n is the dimension of the underlying manifold.

In this paper we establish a formal connection between the average decay of the Fourier transform of functions with respect to a given measure and the Hausdorff behavior of that measure. We also present a generalization of the classical restriction theorem of Stein and Tomas replacing the sphere with sets of prefixed Hausdorff dimension n - 1 + α, with 0 &lt; α &lt; 1.

We provide new assertions on factorization of tent spaces.

Properties of a maximal function for vector-valued martingales were studied by the author in an earlier paper. Restricting here to the dyadic setting, we prove the equivalence between (weighted) $L^p$ inequalities and weak type estimates, and discuss an extension to the case of locally finite Borel measures on ℝⁿ. In addition, to compensate for the lack of an $L^{\infty }$ inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.

This paper deals with function spaces of varying smoothness $B_p^{,s₀}\left(ℝⁿ\right)$, where the function :x ↦ s(x) determines the smoothness pointwise. Those spaces were defined in [2] and treated also in [3]. Here we prove results about interpolation, trace properties and present a characterization of these spaces based on differences.

Some martingale analogues of Sawyer's two-weight norm inequality for the Hardy-Littlewood maximal function Mf are shown for the Doob maximal function of martingales.

We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c₀.

We characterize the pairs of weights on ℝ for which the operators $M_{h,k}^{+}f\left(x\right)=su{p}_{c>x}h(x,c){ʃ}_x^{c}f\left(s\right)k(x,s,c)ds$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on $(x,c):x<c$, while k is defined on $(x,s,c):x<s<c$. If $h(x,c)={(c-x)}^{-\beta }$, $k(x,s,c)={(c-s)}^{\alpha -1}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator $M_{\alpha ,\beta }^{+}f=su{p}_{c>x}1/{(c-x)}^{\beta }{ʃ}_x^{c}f\left(s\right)/{(c-s)}^{1-\alpha }ds$. For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal weights and...

We study the boundedness of the one-sided operator $g{⁺}_{\lambda ,\phi }$ between the weighted spaces $L^p\left(M¯w\right)$ and $L^p\left(w\right)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g{⁺}_{\lambda ,\phi }$. For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g{⁺}_{\lambda ,\phi }$ from $L^p\left({\left(M¯\right)}^{[p/2]+1}w\right)$ to $L^p\left(w\right)$, where ${\left(M¯\right)}^k$ denotes the operator M¯ iterated k times.