Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

In this paper, the authors establish the two-weight boundedness of the local fractional maximal operators and local fractional integrals on Gaussian measure spaces associated with the local weights. More precisely, the authors first obtain the two-weight weak-type estimate for the local-$a$ fractional maximal operators of order $\alpha$ from $L^{p}(v)$ to $L^{q,\infty}(u)$ with 1\leq p\leq qa} A_{p,q,\alpha}^{b'}, and then obtain the two-weight weak-type estimate for the local fractional integrals. In addition, the authors obtain the two-weight strong-type boundedness of the local fractional maximal operators under a condition of $(u,v)\in\mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ and the two-weight strong-type boundedness of the local fractional integrals. These estimates are established by the radialization method and dyadic approach.Comment: 25 pages, 2 figure

Local generation of hydrogen for enhanced photothermal therapy.

By delivering the concept of clean hydrogen energy and green catalysis to the biomedical field, engineering of hydrogen-generating nanomaterials for treatment of major diseases holds great promise. Leveraging virtue of versatile abilities of Pd hydride nanomaterials in high/stable hydrogen storage, self-catalytic hydrogenation, near-infrared (NIR) light absorption and photothermal conversion, here we utilize the cubic PdH0.2 nanocrystals for tumour-targeted and photoacoustic imaging (PAI)-guided hydrogenothermal therapy of cancer. The synthesized PdH0.2 nanocrystals have exhibited high intratumoural accumulation capability, clear NIR-controlled hydrogen release behaviours, NIR-enhanced self-catalysis bio-reductivity, high NIR-photothermal effect and PAI performance. With these unique properties of PdH0.2 nanocrystals, synergetic hydrogenothermal therapy with limited systematic toxicity has been achieved by tumour-targeted delivery and PAI-guided NIR-controlled release of bio-reductive hydrogen as well as generation of heat. This hydrogenothermal approach has presented a cancer-selective strategy for synergistic cancer treatment