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中国科学院大学数学科学学院, 北京 100049
摘要: 经典的欧氏空间中的卷积如下给出。对
关键词: 卷积闭超曲面有界性
The classical concept of convolution operator has been generalized in many new cases. The reason is that convolution operator has many applications in harmonic analysis and engineering. For example, it can be used to characterize the bounded operators which commutate with transition actions.
Many researchers have made explorations in these topics. For instance, Oinarov[1] explored the boundedness and compactness of convolution operators of fractional integration type. Avsyankin[2] and Guliyeva and Sadigova[3] explored the properties of convolution operators on Morrey spaces.
Harmonic analysis on Euclidean space has developed very fast. It is also meaningful to generalize the theories on manifolds. For example, the progress of restriction conjecture about Fourier transformation has been introduced in Refs. [4-7]. Similarly, we consider the restriction properties of convolution integral on manifolds in this study.
1 Some definitionsBefore we put forward our main results, some useful definitions are given as follows.
Definition 1.1??Sobolev space
$W_1^k\left( {{\mathbb{R}^n}} \right) = \left\{ {f \in {L^1}\left( {{\mathbb{R}^n}} \right):{\partial ^\alpha }f \in {L^1}\left( {{\mathbb{R}^n}} \right);|\alpha | \leqslant k} \right\}.$ |
${\left\| f \right\|_{W_1^k}} = {\left\| f \right\|_1} + \sum\limits_{\left| {{\alpha _\alpha }} \right| \le k} {{{\left\| {{\partial ^\alpha }f} \right\|}_1}} $ |
$\left\| {{T_f}(g)} \right\|_{p,M}^p: = \int_M {{{\left| {{T_f}(g)(x)} \right|}^p}} {\rm{d}}\sigma (x),$ | (1) |
2 Main resultWe state our main theorem as follows.
Theorem 2.1??Let M be a closed (connected compact without boundary) (n-1) dimensional hypersurface in
${\left\| {{T_f}(g)} \right\|_{p,M}} \le {C^\prime }(M){\left\| f \right\|_{w_1^1}}{\left\| g \right\|_p}$ |
3 Proof of the main resultAccording to Lemma 3.1(see below), we only need to prove Theorem 2.1 for all
Lemma 3.1??Let M be a closed (n-1) dimensional hypersurface in
${\left\| {{T_f}(g)} \right\|_{p,M}} \le C(M){\left\| f \right\|_{W_1^1}}{\left\| g \right\|_p}$ | (2) |
Proof:
In fact, under the hypothesis in the lemma, given any
${\left\| {{f_k} - f} \right\|_{W_1^1}} \to 0,$ | (3) |
${\left\| {{g_k} - g} \right\|_p} \to 0.$ | (4) |
Thus, applying Fatou's lemma and using (2), (3), and (4), we have
$\begin{gathered} {\left\| {{T_f}(g)} \right\|_{p,M}} = {\left\| {f * g} \right\|_{p,M}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leqslant {\left\| {|f|*|g|} \right\|_{p,M}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\left\| {\int_{{\mathbb{R}^n}} {\mathop {\lim }\limits_{k \to \infty } {{\left| f \right|}_k}\left( {x - y} \right){{\left| g \right|}_k}\left( y \right){\text{d}}y} } \right\|_{p,M}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leqslant \mathop {\lim \inf }\limits_{k \to \infty } {\left\| {\left| {{f_k}} \right|*\left| {{g_k}} \right|} \right\|_{p,M}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leqslant \mathop {\lim \inf }\limits_{k \to \infty } C\left( M \right){\left\| {{f_k}} \right\|_{W_1^1}}{\left\| {{g_k}} \right\|_p} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = C\left( M \right){\left\| f \right\|_{W_1^1}}{\left\| g \right\|_p}. \hfill \\ \end{gathered} $ |
Then, we state the following tubular neighborhood lemma[8].
Lemma 3.2??Let S be a closed hypersurface in Euclidean space. (N, S, π,
Then, because M is a closed hypersurface in
Lemma 3.3??Let M be a (n-1) dimensional closed hypersurface in
${\left\| {{T_f}(g)} \right\|_{p,M}} \le C(M){\left\| f \right\|_{W_1^1}}{\left\| g \right\|_p}$ |
Proof of Lemma 3.3 and Theorem 2.1:
We first prove that the inequality holds for 1 < p < ∞. Since f(x-y)g(y)=(f1(x-y)+if2(x-y))(g1(y)+ig2(y)), we are able to suppose f and g are real valued functions.
Let Ω be the volume form on M, and let i:
$\begin{array}{*{20}{c}}{\mathit{\Omega } = {i^*}\sum\limits_{\alpha = 1}^n {{{\left( { - 1} \right)}^{\alpha - 1}}} {\eta ^\alpha }(x){\rm{d}}{x^1} \wedge \cdots }\\{ \wedge \widehat {{\rm{d}}{x^\alpha }} \wedge \cdots \wedge {\rm{d}}{x^n},}\end{array}$ | (5) |
Therefore, substituting surface measure dσ in (1) by volume form Ω in (5), we obtain
$\begin{array}{l}\left\| {{T_f}(g)} \right\|_{p,M}^p = \int_M {{{\left| {{T_f}(g)(x)} \right|}^p}} {\rm{d}}\sigma (x)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \int_M {{{\left| {{T_f}(g)(x)} \right|}^p}} \Omega = \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{\alpha = 1}^n {\int_M {{i^*}} } {h_0}{( - 1)^{\alpha - 1}}{\eta ^\alpha }{\rm{d}}{x^1} \wedge \cdots \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\widehat {{\rm{d}}{x^\alpha }} \cdots \wedge {\rm{d}}{x^n},\end{array}$ | (6) |
Since, for p>1, h0(x) is smooth for ε>0 by Sard theorem[9], there exists a
$\begin{array}{l}\left\| {{T_f}(g)} \right\|_{p,M}^p = \int_M {{{\left| {{T_f}(g)(x) - c + c} \right|}^p}} {\rm{d}}\sigma (x)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le {2^{p - 1}}{\varepsilon ^p}|M| + {2^{p - 1}}\int_M h (x){\rm{d}}\sigma (x).\end{array}$ | (7) |
${(a + b)^p} \le {2^{p - 1}}\left( {{a^p} + {b^p}} \right).$ | (8) |
$\int_{{\Gamma _\varepsilon }} {\left| {\frac{{\partial h}}{{\partial {x^\alpha }}}} \right|} {\rm{d}}x < \varepsilon \cdot (\alpha \in \{ 1, \cdots ,n\} ).$ | (9) |
$\begin{array}{l}\left| {\int_M h (x){\rm{d}}\sigma (x)} \right|\\ = \left| {\int_M {{i^*}} h{{( - 1)}^{\alpha - 1}}{\eta ^\alpha }{\rm{d}}{x^1} \wedge \cdots \wedge \widehat {{\rm{d}}{x^\alpha }} \wedge \cdots \wedge {\rm{d}}{x^n}} \right|\\ = \left| {\int_D {\frac{{\partial h}}{{\partial {x^\alpha }}}} {\eta ^\alpha }{\rm{d}}x + \int_D h \frac{{\partial {\eta ^\alpha }}}{{\partial {x^\alpha }}}{\rm{d}}x} \right|\\ \le C(M)\left( {\int_D {\left| {\frac{{\partial h}}{{\partial {x^\alpha }}}} \right|{\rm{d}}x} + \int_D {\left| h \right|{\rm{d}}x} } \right),\end{array}$ | (10) |
Next, we estimate the two parts in (10) separately. Using (9), we first have
$\begin{array}{l}\int_D {\left| {\frac{{\partial h}}{{\partial {x^\alpha }}}} \right|} {\rm{d}}x = \left( {\int_{D - {\Gamma _\varepsilon }} {} + \int_{{\Gamma _\varepsilon }} {} } \right)\left| {\frac{{\partial h}}{{\partial {x^\alpha }}}} \right|{\rm{d}}x\\ \le \varepsilon + \int_{D - {\Gamma _\varepsilon }} {\left| {\frac{{\partial h}}{{\partial {x^\alpha }}}} \right|} {\rm{d}}x\\ = \varepsilon + \int_{D - {\Gamma _\varepsilon }} {\left| {\frac{\partial }{{\partial {x^\alpha }}}} \right|} {T_f}(g) - c\left| {^p} \right|{\rm{d}}x\\ = \int_{D - {\Gamma _\varepsilon }} p {\left| {{T_f}(g) - c} \right|^{p - 1}}\left| {\frac{{\partial \left| {{T_f}(g) - c} \right|}}{{\partial {x^\alpha }}}} \right|{\rm{d}}x + \varepsilon .\end{array}$ | (11) |
$\begin{array}{l}\int_D {\left| {\frac{{\partial h}}{{\partial {x^\alpha }}}} \right|} {\rm{d}}x \le \varepsilon + \int_{D - {\Gamma _\varepsilon }} p {\left| {{T_f}(g) - c} \right|^{p - 1}}\left| {{T_{\frac{{\partial {\rm{f}}}}{{\partial {x^\alpha }}}}}(g)} \right|{\rm{d}}x\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \varepsilon + \int_D p {\left| {{T_f}(g) - c} \right|^{p - 1}}\left| {{T_{\frac{{\partial f}}{{\partial {x^\alpha }}}}}(g)} \right|{\rm{d}}x\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \varepsilon + p\left\| {{T_f}(g) - c} \right\|_{p,D}^{p - 1}{\left\| {{T_{\frac{{\partial f}}{{\partial {x^\alpha }}}}}(g)} \right\|_{p,D}}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \varepsilon + {2^{{{(p - 1)}^2}}}p|D{|^{\frac{{p - 1}}{p}}}{\varepsilon ^{p - 1}}{\left\| {{T_{\frac{{\partial f}}{{\partial {x^\alpha }}}}}(g)} \right\|_p} + \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{2^{{{(p - 1)}^2}}}p\left\| {{T_f}(g)} \right\|_p^{p - 1}{\left\| {{T_{\frac{{\partial f}}{{\partial {x^\alpha }}}}}(g)} \right\|_p}.\end{array}$ | (12) |
$\begin{array}{*{20}{c}}{{{\left\| {{T_f}(g)} \right\|}_p} \le {{\left\| f \right\|}_1}{{\left\| g \right\|}_p},}\\{{{\left\| {{T_{\frac{{\partial f}}{{\partial {x^\alpha }}}}}(g)} \right\|}_p} \le {{\left\| {\frac{{\partial f}}{{\partial {x^\alpha }}}} \right\|}_1}{{\left\| g \right\|}_p},}\end{array}$ |
$\begin{array}{l}\int_D {\left| h \right|{\rm{d}}x} \le {2^{p - 1}}\left( {\int_D {{{\left| {{T_f}\left( g \right)\left( x \right)} \right|}^p}{\rm{d}}x} + {\varepsilon ^p}\left| D \right|} \right)\\\;\;\;\;\;\;\;\;\;\;\; \le {2^{p - 1}}{\varepsilon ^p}\left| D \right| + {2^{p - 1}}\left\| f \right\|_1^p\left\| f \right\|_p^p,\end{array}$ | (13) |
$\begin{array}{l}\int_D {\left| {\frac{{\partial h}}{{\partial {x^\alpha }}}} \right|} {\rm{d}}x \le \varepsilon + {2^{(p - 1)2}}p|D{|^{\frac{{p - 1}}{p}}}{\varepsilon ^{p - 1}}{\left\| {{T_{\frac{{\partial f}}{{\partial {x^\alpha }}}}}(g)} \right\|_p} + \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;{2^{{{(p - 1)}^2}}}p\left\| g \right\|_p^p\left\| f \right\|_1^{p - 1}{\left\| {\frac{{\partial f}}{{\partial {x^\alpha }}}} \right\|_1}.\end{array}$ | (14) |
$\begin{array}{l}\left\| {{T_f}(g)} \right\|_{p,M}^p \le {2^{p - 1}}{\varepsilon ^p}|M| + {2^{p - 1}}\int_M h (x){\rm{d}}\sigma (x)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le {2^{p - 1}}{\varepsilon ^p}|M| + {2^{p - 1}}C(M)\left( {{2^{p - 1}}{\varepsilon ^p}|D| + \varepsilon + } \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left( {{2^{p - 1}} + {2^{{{(p - 1)}^2}}}p} \right)\left\| f \right\|_1^{p - 1}{{\left\| f \right\|}_{W_1^1}}\left\| g \right\|_p^p} \right)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le {2^{p - 1}}{\varepsilon ^p}|M| + {2^{p - 1}}C(M)\left( {{2^{p - 1}}{\varepsilon ^p}|D| + \varepsilon + } \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{array}{*{20}{l}}{\left. {{2^{{{(p - 1)}^2}}}p|D{|^{\frac{{p - 1}}{p}}}{\varepsilon ^{p - 1}}{{\left\| {{T_{\frac{{\partial f}}{{\partial {x^\alpha }}}}}(g)} \right\|}_p}} \right) + }\\{{2^{p - 1}}\left( {{2^{p - 1}} + {2^{{{(p - 1)}^2}}}p} \right)C(M)\left\| f \right\|_{W_1^1}^{{p_1}}\left\| g \right\|_p^p.}\end{array}\end{array}$ | (15) |
$\left\| {{T_f}(g)} \right\|_{p,M}^p \le C(p,M)\left\| f \right\|_{W_1^1}^p\left\| g \right\|_p^p.$ | (16) |
For p=1, we choose
${\left\| {{T_f}(g)} \right\|_{1,M}} \le 2C(M){\left\| f \right\|_{W_1^1}}{\left\| g \right\|_1}.$ | (17) |
Meanwhile, it is obvious that
${\left\| {{T_f}(g)} \right\|_{\infty ,M}} \le {\left\| f \right\|_1}{\left\| g \right\|_\infty }$ | (18) |
Finally, using (17), (18), and Riesz-Th?rin interpolation theorem[11], we can choose a constant C′(M)=max{2C(M), 1} which is independent of p such that the following is true.
${\left\| {{T_f}(g)} \right\|_{p,M}} \le C'(M){\left\| f \right\|_{W_1^1}}{\left\| g \right\|_p}.$ |
References
[1] | Oinarov R. Boundedness and compactness of a class of convolution integral operators of fractional integration type[J]. Proceedings of the Steklov Institute of Mathematics, 2016, 293(1): 255-271. DOI:10.1134/S0081543816040180 |
[2] | Avsyankin O G. On the compactness of convolution-type operators in Morrey spaces[J]. Mathematical Notes, 2017, 102(3/4): 437-443. |
[3] | Guliyeva F A, Sadigova S R. On some properties of convolution in morrey type spaces[J]. Azerbaijan Journal of Mathematics, 2017, 8(1): 140-150. |
[4] | Tomas P A. A restriction theorem for the Fourier transform[J]. Bulletin of the American Mathematical Society, 1975, 81(1991): 31-36. |
[5] | Wolff T H. Lectures on harmonic analysis[M]. Rhode Island: American Mathematical Society, 2002. |
[6] | Stein E M. Harmonic analysis, real variable methods, orthogonality, and osciallatory integrals[M]. New Jersey: Princeton University Press, 1993. |
[7] | Tao T. Recent progress on the restriction conjecture[J]. Mathematics, 2003, 217-243. |
[8] | Zhang Z S. Lecture notes on differential topology[M]. Beijing: Peking University Press, 2002. |
[9] | Spivak M. A comprehensive introduction to differential geometry volume Ⅰ[M]. 3rd ed. Houston: Publish or Perish, 1999. |
[10] | Spivak M. A comprehensive introduction to differential geometry volume Ⅱ[M]. 3rd ed. Houston: Publish or Perish, 1999. |
[11] | Stein E M. Singular integerals and differentiability properrities of functions[M]. New Jersey: Princeton University Press, 1970. |