臺灣博碩士論文加值系統

在此篇論文中我們推廣在四個不同區段: (i) 1 < p < d/α
(ii) p=d/α(iii) d/α<p<d/α-1 (iv) p = d/α-1
裡的分數積分不等式。相對於使用Riesz 核，
我們的估計在更一般的Lorentz 空間上使用Bessel 核。此外，我們在Lorentz 空
間上給出了一個新的Bessel 核之超臨界估計:
另1 < p < d/α-1 且 1 <= q <= ∞。如果f ∈ Lp;q(R^d)，那麼存在常數C = C(α; d; p; q)
使得
|g_α*f(x) - g_α*f(z)| <= C|x-z|(|ln(|x-z|)|+1)^(1/q') ∥f∥Lp;q.

In this thesis we extend some results concerning fractional integration inequalities in four different regimes: (i) 1 < p < d/α (ii) p=d/α(iii) d/α<p<d/α-1 (iv) p = d/α-1
In contrast to the results where the Riesz kernel is utilized, we have treated the Bessel kernel, while our estimates are on the more general scale of Lorentz space. In particular, we establish a new estimate with Bessel potentials on Lorentz space in the supercritical exponent:
Let 1 < p < d/α-1 and 1 <= q <= ∞. If f ∈ Lp;q(R^d), then there exists constant C = C(α; d; p; q) such that
|g_α*f(x) - g_α*f(z)| <= C|x-z|(|ln(|x-z|)|+1)^(1/q') ∥f∥Lp;q.

Ch1 Introduction 1

Ch2 Preliminaries p6
2.1 Convolution and Lorentz Spaces . . . . . . . . . p6
2.2 Definition of Bessel Kernels . . . . . . . . . . p10

Ch3 Technical Lemma p11

ch4 Estimate Below The Supercritical Exponent p28
4.1 Lorentz Space Estimate . . . . . . . . . . . . . p28
4.2 Exponential Inequality Estimate . . . . . . . . . p29
4.3 Hölder Continuity Estimate . . . . . . . .. . . . p31

Ch5 The Supercritical Estimate p33

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