Volume 7, Issue 2

We prove a Riesz–Herz estimate for the maximal function MCμf(x)=supx∈BC(B)-1∫B|f(y)|dμ(y)${M_{C}^{\mu } f(x) = \sup _{x \in B} C(B)^{-1} \int _{B} \vert f(y)\vert \, d \mu (y)}$ associated to a capacity C on a metric measure space ( X , d ,μ). This estimate extends the equivalence (Mf)*(t)≃f**(t)${(M f)^{*}(t) \simeq f^{**}(t)}$ for the usual Hardy–Littlewood maximal function M f and the Riesz–Herz estimate for the capacitary maximal function on ℝ n . Essential tools are the extension of the Wiener–Stein estimate for the distribution function of MCμ${M_C^\mu }$ and the existence of appropriate dyadic cubes in metric measure spaces. Finally, we obtain the Riesz–Herz estimate for a discrete version of the capacitary maximal function.

Let G be a group with identity e . Let R be a G -graded commutative ring and M a graded R -module. In this paper, we introduce the concept of graded classical primary submodules. Various properties of graded classical primary submodules are considered.

In this paper, we show the existence and multiplicity of nontrivial non-negative solutions of the fractional p -Kirchhoff problem M∫ℝ2n|u(x)-u(y)|p|x-y|n+psdxdy(-Δ)psu=λf(x)|u|q-2u+g(x)|u|r-2uinΩ,u=0inℝn∖Ω,$ M\Bigg (\int _{\mathbb {R}^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\,dx\,dy\Biggr )(-\Delta )^{s}_p u =\lambda f(x)|u|^{q-2}u+ g(x)|u|^{r-2}u\quad \mbox{in } \Omega ,\qquad u=0\quad \mbox{in }\mathbb {R}^{n}\setminus \Omega , $ where (-Δ) p s is the fractional p -Laplace operator, Ω is a bounded domain in ℝ n with smooth boundary, f ∈ L r /( r - q ) (Ω) and g ∈ L ∞ (Ω) are sign changing, and M is a continuous function. Moreover, p s < n < 2 p s and 1 < q < p < r ≤ p s * = n p /( n - p s ).

In this paper, we study some properties of transversal hypersurfaces of a nearly trans-Sasakian manifold. Furthermore, we give some characterizations of totally geodesic regarding such manifolds. We also prove that if M is a transversal hypersurface of a nearly cosymplectic manifold M ˜, equipped with a contact structure under the condition that a vector field V is parallel, then M is totally geodesic.

We study the Dirichlet problem for Hamilton–Jacobi equations of the form H ( D u ( x )) - a ( x ) = 0 in Ω, u ( x ) = ϕ( x ) on ∂Ω, under suitable hypotheses on the function H and without continuity assumptions on the map a . We find a class of maps a contained in the space L ∞ (Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution.

In this paper, we give necessary and sufficient conditions for a homogeneous cone Ω to be symmetric in two ways. One is by using the multiplier matrix of Ω, and the other is in terms of the basic relative invariants of Ω. In the latter approach, we need to show that the real parts of certain meromorphic rational functions obtained by the basic relative invariants are always positive on the tube domains over Ω. This is a generalization of a result of Ishi and Nomura [Math. Z. 259 (2008), 604–674].