We study L p → L r estimates for restricted averaging operators related to algebraic varieties V of d-dimensional vector spaces over finite fields Fq with q elements. We observe properties of both the Fourier restriction operator and the averaging operator over V ⊂ F d q . As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions d ≥ 2, and cones in odd dimensions d ≥ 3. In addition, when the variety V is a cone lying in an even dimensional vector space over Fq and −1 is a square number in Fq, we also obtain sharp estimates except for two endpoints.
We investigate the sharp L p → L r estimates for the restricted averaging operator A C over the cone C of the d-dimensional vector space F d q over the finite field Fq with q elements. The restricted averaging operator A C for the cone C is defined by the relation that A C f = f * σ| C , where σ denotes the normalized surface measure on the cone C, and f is a complex valued function on the space F d q with the normalized counting measure dx. In the previous work [15], the sharp boundedness of A C was obtained in odd dimensions d ≥ 3 but partial results were only given in even dimensions d ≥ 4. In this paper we prove the optimal estimates in even dimensions d ≥ 6 in the case when the cone C ⊂ F d q contains a d/2 dimensional subspace.
The first two authors contributed equally to this work Although several studies have shown physiological functions of interleukin (IL)-32, the role of IL-32 in liver has not yet been reported. This study was initiated to examine the effects of IL-32y on APAPinduced acute hepatic failure in IL-32y transgenic mice. IL-32y overexpressing and non-transgenic mice received 500 mg/kg Acetoaminophen (APAP) intraperitoneally. Serum alanine transaminase and aspartate transaminase were significantly lower in the APAP treated IL-32y overexpressing mice compared with those APAP-treated non-transgenic. IL-32y markedly reduced a restricted area of the necrosis and inflammation. APAP-induced reduced glutathione depletion, induction of nitric oxide and lipid peroxidation, and cytochrome P4502El expression was significantly lowered in the IL-32y overexpressing mice. Elevation of Kupffer cells and natural killer cells by APAP were reduced in the IL-32y overexpressing mice. The expression ofIL-la, IL-lra, macrophage inflammatory protein-2, C-C motif chemokine ligand 5 and tissue inhibitor of metalloproteinase-l was increased by APAP in nontransgenic mice, and were lowered in the IL-32y overexpressing mice. Moreover, APAP-induced nuclear transcription factor-kappa B (NF-KB) and signal transducers and activators oftranscription 1 (STATl) activities were greatly lowered in the IL-32y overexpressing mice. The results indicate that IL-32y could effectively inhibit drug-induced hepatic failure, and reduce the number of cytotoxic immune cells and pro-inflammatory cytokine production through reduced activities of NF-KB and STATl. This might be attributable to lowering APAP-induced liver toxicity in IL-32y overexpressing mice.
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