The Journal of Geometric Analysis最新文献

In this short note, we give an easy proof of the following result: for ( nge 2, ) (underset{trightarrow 0}{lim } ,e^{itDelta }fleft( x+gamma (t)right) = f(x) ) almost everywhere whenever ( gamma ) is an ( alpha )-Hölder curve with ( frac{1}{2}le alpha le 1 ) and ( fin H^s({mathbb {R}}^n) ), with ( s > frac{n}{2(n+1)} ). This is the optimal range of regularity up to the endpoint.

In this paper, our object of investigation is the commutators of the Stein’s square functions asssoicated with the Bochner-Riesz means of order ({uplambda }) defined by

where (widehat{K_t^{uplambda }}({upxi })=frac{|{upxi }|^2}{t^2}Big (1-frac{|{upxi }|^2}{t^2}Big )_+^{{uplambda }-1}) and (bin mathrm BMO(mathbb {R}^n)). We show that (G_{b,m}^{uplambda }) is a compact operator from (L^p(w)) to (L^p(w)) for (1<p<infty ) and ({uplambda }>frac{n+1}{2}) whenever (bin mathrm CMO({mathbb {R}^n})), where (textrm{CMO}(mathbb {R}^n)) is the closure of (mathcal {C}_c^infty (mathbb {R}^n)) in the (textrm{BMO}(mathbb {R}^n)) topology.

For an Orlicz function (varphi ) with critical lower type (i(varphi )in (0, 1)) and upper type (I(varphi )in (0,1)), set (m(varphi )=lfloor n(1/i(varphi )-1)rfloor ). In this paper, the authors establish bilinear decomposition for the product of the Orlicz–Hardy space (H^{varphi }({mathbb {R}}^{n})) and its dual space—the Orlicz–Campanato space ({mathfrak {L}}_{varphi }({mathbb {R}}^{n})). In particular, the authors prove that the product (in the sense of distributions) of (fin H^{varphi }({mathbb {R}}^{n})) and (gin {mathfrak {L}}_{varphi }({mathbb {R}}^{n})) can be decomposed into the sum of S(f, g) and T(f, g), where S is a bilinear operator bounded from (H^{varphi }({mathbb {R}}^{n})times {mathfrak {L}}_{varphi }({mathbb {R}}^{n})) to (L^{1}({mathbb {R}}^{n})) and T is another bilinear operator bounded from (H^{varphi }({mathbb {R}}^{n})times {mathfrak {L}}_{varphi }({mathbb {R}}^{n})) to the Musielak–Orlicz–Hardy space (H^{Phi }({mathbb {R}}^{n})), with (Phi ) being a Musielak–Orlicz function determined by (varphi ). The bilinear decomposition is sharp in the following sense: any vector space ({mathcal {Y}}subset H^{Phi }({mathbb {R}}^{n})) that adapted to the above bilinear decomposition should satisfy ( L^infty ({mathbb {R}}^{n})cap {mathcal {Y}}^{*}=L^infty ({mathbb {R}}^{n})cap (H^{Phi }({mathbb {R}}^{n}))^{*} ). Indeed, (L^infty ({mathbb {R}}^{n})cap (H^{Phi }({mathbb {R}}^{n}))^{*}) is just the multiplier space of ({mathfrak {L}}_{varphi }({mathbb {R}}^{n})). As applications, the authors obtain not only a priori estimate of the div-curl product involving the space (H^{Phi }({mathbb {R}}^{n})), but also the boundedness of the Calderón–Zygmund commutator [b, T] from the Hardy type space (H^{varphi }_{b}({mathbb {R}}^{n})) to (L^{1}({mathbb {R}}^{n})) or (H^{1}({mathbb {R}}^{n})) under (bin {mathfrak {L}}_{varphi }({mathbb {R}}^{n})), (m(varphi )=0) and suitable cancellation conditions of T.

This paper is concerned with the existence of normalized solutions to a mass-supercritical quasilinear Schrödinger equation:

satisfying the constraint (int _{{mathbb {R}}^N}u^2=a). We will investigate how the potential and the nonlinearity effect the existence of the normalized solution. As a consequence, under a smallness assumption on V(x) and a relatively strict growth condition on g, we obtain a normalized solution for (N=2), 3. Moreover, when V(x) is not too small in some sense, we show the existence of a normalized solution for (Nge 2) and (g(u)={u}^{q-2}u) with (4+frac{4}{N}<q<2cdot 2^*).

In Palmer and Pámpano (Calc Var Partial Differ Equ 61:79, 2022), the authors studied a particular class of equilibrium solutions of the Helfrich energy which satisfy a second order condition called the reduced membrane equation. In this paper we develop and apply a second variation formula for the Helfrich energy for this class of surfaces. The reduced membrane equation also arises as the Euler–Lagrange equation for the area of surfaces under the action of gravity in the three dimensional hyperbolic space. We study the second variation of this functional for a particular example.

In this paper, we study the (L^p({mathbb {R}}^2))-improving bounds, i.e., (L^p({mathbb {R}}^2)rightarrow L^q({mathbb {R}}^2)) estimates, of the maximal function (M_{gamma }) along a plane curve ((t,gamma (t))), where

and (gamma ) is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain (M_{gamma }: L^p({mathbb {R}}^2)rightarrow L^q({mathbb {R}}^2)) if (left( frac{1}{p},frac{1}{q}right) in Delta cup {(0,0)}) and (left( frac{1}{p},frac{1}{q}right) ) satisfying (1+(1 +omega )left( frac{1}{q}-frac{1}{p}right) >0), where (Delta :=left{ left( frac{1}{p},frac{1}{q}right) : frac{1}{2p}<frac{1}{q}le frac{1}{p}, frac{1}{q}>frac{3}{p}-1 right} ) and (omega :=limsup _{trightarrow 0^{+}}frac{ln |gamma (t)|}{ln t}). This result is sharp except for some borderline cases.

We prove a Li-Yau type eigenvalue-diameter estimate for signed graphs. That is, the nonzero eigenvalues of the Laplacian of a non-negatively curved signed graph are lower bounded by (1/D^2) up to a constant, where D stands for the diameter. This leads to several interesting applications, including a volume estimate for non-negatively curved signed graphs in terms of frustration index and diameter, and a two-sided Li-Yau estimate for triangle-free graphs. Our proof is built upon a combination of Chung-Lin-Yau type gradient estimate and a new trick involving strong nodal domain walks of signed graphs. We further discuss extensions of part of our results to nonlinear Laplacians on signed graphs.

We prove that there exists a gradient expanding Ricci soliton asymptotic to any given cone over the product of a round sphere and a Ricci flat manifold. In particular we obtain asymptotically conical expanding Ricci solitons with positive scalar curvature on (mathbb {R}^3 times S^1.) More generally we construct continuous families of gradient expanding Ricci solitons on trivial vector bundles over products of Einstein manifolds with arbitrary Einstein constants.

Let X be a reduced complex space of pure dimension. We consider divergent integrals of certain forms on X that are singular along a subvariety defined by the zero set of a holomorphic section of some holomorphic vector bundle (E rightarrow X). Given a choice of Hermitian metric on E we define a finite part of the divergent integral. Our main result is an explicit formula for the dependence on the choice of metric of the finite part.

Let ((D, pi )) be an unramified Riemann domain over a Stein manifold of dimension n. Assume that (H^k(D,mathscr {O}) = 0) for (2 le k le n - 1) and there exists a complex Lie group G of positive dimension such that all differentiably trivial holomorphic principal G-bundles on D are holomorphically trivial. Then, we prove that D is Stein.