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MATHEMATICS
I. B. SIMONENKO
BOUNDEDNESS OF SINGULAR INTEGRALS IN ORLICZ SPACES
(Presented by Academician S. L. Sobolev on 23 X 1959)
Let \(\mathscr L_M^*(D)\) be the Orlicz space defined by a bounded measurable set \(D\) of \(m\)-dimensional space \(E_m\) and by the function
\[ M(u)=\int_0^{|u|} p(t)\,dt, \]
where \(p(t)\) is a nondecreasing function;
\[ N(u)=\int_0^{|u|} q(t)\,dt, \]
where \(q(t)\) is the right inverse of \(p(t)\), and let \(\mathscr L_N(D)\) be the set of functions \(V\) for which
\[ \int_D N[v(P)]\,dP \]
exists.
As is known ((\(^{4}\), p. 83)), the space \(\mathscr L_M^*(D)\) consists of functions \(u\) satisfying the condition
\[ \int_D |u|\,|v|\,dP<\infty \]
for all \(v\in \mathscr L_N(D)\); the norm in the space \(\mathscr L_M^*(D)\) is introduced by the equality
\[ \|u\|_M=\sup_v \int_D u(P)v(P)\,dP,\qquad \int_D N[v(P)]\,dP\leqslant 1. \]
Consider the singular integral
\[ Kf=\int_D \frac{\Omega(P;\theta)}{|P-Q|^m}\,f(Q)\,dQ, \tag{1} \]
where
\[ \theta=\frac{P-Q}{|P-Q|} \]
is a point of the unit sphere.
\(\Omega(P;\theta)\) satisfies the following conditions:
-
\[ \int_{S_1}\Omega(P;\theta)\,ds_\theta=0, \]
where \(S_1\) is the unit sphere. -
\(\Omega(P;\theta)\) is continuous in \(\theta\) for fixed \(P\).
-
\[ |\Omega(P;\theta_1)-\Omega(P;\theta_2)|\leqslant \omega(|\theta_1-\theta_2|), \]
where \(\omega\) does not depend on \(P\) and satisfies condition 4. -
\[ \int_0^1 \frac{\omega(t)}{t}\,dt<\infty. \]
-
\[ |\Omega(P_1;\theta)-\Omega(P_2;\theta)|\leqslant B|P_1-P_2|^\alpha;\quad \alpha>0; \]
\(B\) is a constant.
We subject the function \(M(u)\) to the additional condition: for sufficiently large \(u\) the inequality
\[ 1<\beta\leqslant \frac{u p(u)}{M(u)}\leqslant \alpha \tag{2} \]
holds.
It is easy to prove that there exists a function \(M_1\), equivalent to the function \(M\) ((\(^{4}\), p. 27)), satisfying inequality (2) for all \(u\). At the same time ...
\(\mathscr L^*_{M_1}=\mathscr L^*_M(D)\), and the norms \(\|\ \|_{M_1}\) and \(\|\ \|_M\) are equivalent \((({}^4),\) p. 130). Therefore, without loss of generality, we may assume that inequality (2) holds for all \(u\).
Theorems on the boundedness of singular integrals in the class \(\mathscr L_p\) (\(p>1\)) were obtained by S. G. Mikhlin \(({}^1,{}^3)\) and by A. Calderón and A. Zygmund \(({}^2)\). The space \(\mathscr L_p\) is a special case of \(\mathscr L^*_M(D)\), when \(\alpha=\beta=p\). We have obtained a theorem generalizing the results of the authors mentioned.
Theorem. The singular operator (1) is defined and bounded in the space \(\mathscr L^*_M(D)\), i.e.
\[ \|Kf\|_M \leq C\|f\|_M, \tag{3} \]
where \(C\) depends only on \(\alpha,\beta,\Omega,u_0,D\) *.
Proof. Introduce the notation:
\[ K_\lambda(P;Q)= \begin{cases} \dfrac{\Omega(P;\theta)}{|P-Q|^m}, & \text{if } |P-Q|>\lambda,\\ 0, & \text{if } |P-Q|\leq \lambda; \end{cases} \]
\[ |f(P)|_y= \begin{cases} f(P), & \text{if } f(P)\leq y,\\ y, & \text{if } f(P)>y; \end{cases} \]
\(f^*(t)\) is the nonincreasing function equimeasurable ** with \(f(Q)\), defined on the ray \(0\leq t<\infty\); \(\beta_f(x)=\dfrac1x\int_0^x f^*(t)\,dt\); \(\beta f(y)\) is the inverse function of \(\beta_f(x)\); \(\widetilde f_\lambda=K_\lambda f=\)
\[ =\int_D K_\lambda(P;Q)f(Q)\,dQ;\quad E^ \lambda_y \text{ is the set of points } P\in D,\text{ where } |\widetilde f_\lambda(P)|\geq y. \]
Then for any \(r>1\) the following generalized inequality of A. Calderón and A. Zygmund \(({}^2)\) holds:
\[ |E^\lambda_y|\leq \frac{C_1}{y^r}\int_D |f(P)|_y^r\,dP + C_1\beta f(y), \tag{4} \]
where \(C_1\) is a constant depending only on \(D,r\), and \(\Omega\).
This inequality was obtained in \(({}^2)\) for the case \(r=2\) and for a characteristic \(\Omega\) depending on \(\theta\). The stated inequality is proved similarly, using the result of S. G. Mikhlin \((({}^1),\) p. 99, Theorem 3).
Since, for \(f\) satisfying the Hölder condition, \(K_\lambda f\) tends uniformly to \(Kf\) as \(\lambda\to0\), the theorem will be proved if we verify that
\[ \|\widetilde f_\lambda\|_M \leq C\|f\|_M, \tag{5} \]
where \(C\) does not depend on \(\lambda\). Obviously, it is sufficient to prove inequality (5) for \(f\geq0\), which we shall assume in what follows.
We give the necessary inequalities (see \(({}^4),\) pp. 39 and 251)
\[ \frac{up(u)}{N[p(u)]}\geq \frac{\alpha}{\alpha-1}\quad \text{for all } u; \tag{6} \]
\[ \|u\|_M \leq 1+\int_D M[\,u(P)\,]\,dP; \tag{7} \]
* When this note had been prepared for publication, we learned of the work \(({}^6)\), in which, under assumptions somewhat different from ours, boundedness is proved, but only in the case when \(\Omega=\Omega(\theta)\).
** \(f_1(t)\) \((0\leq t<\infty)\) and \(f(P)\) \((P\in D)\) are called equimeasurable if
\(mE(a\leq f_1<b)=mE(a\leq f<b)\).
\[ \int_D u(P)v(P)\,dP \leq \|u\|_M\|v\|_M; \tag{8} \]
if \(\|u\|_M \leq 1\), then
\[ \int_D M[u(P)]\,dP \leq \|u\|_M. \tag{9} \]
Let us proceed to estimate \(\|\widetilde f_\lambda\|_M\):
\[ \|\widetilde f_\lambda\|_M \leq 1+\int_D M[\widetilde f_\lambda(P)]\,dP =1+\int_0^\infty |E^\lambda|\,P(y)\,dy. \tag{10} \]
We estimate the last integral using inequality (4):
\[ \int_0^\infty E_y^\lambda\,p(y)\,dy \leq C_1\int_0^\infty \frac{p(y)}{y^r}\,dy \int_D |f(P)|_y^r\,dP + C_1\int_0^\infty \beta_f(y)p(y)\,dy, \tag{11} \]
We consider the integrals on the right-hand side of (11) separately:
\[ I_1=\int_0^\infty \frac{p(y)}{y^r}\,dy \int_D |f(P)|_y^r\,dP = \int_D M[|f(P)|]\,dP + \int_D |f(P)|^r\,dP\int_{f(P)}^\infty \frac{p(y)}{y^r}\,dy. \]
Choosing \(r>\alpha\) so that \(\lim\limits_{y\to\infty}\dfrac{M(y)}{y^r}=0\) ((4), p. 38), and integrating by parts, we obtain
\[ \int_D |f(P)|^r\,dP\int_{f(P)}^\infty \frac{p(y)}{y^r}\,dy = -\int_D M[|f(P)|]\,dP + r\int_D |f(P)|^r\,dP\int_{f(P)}^\infty \frac{M(y)}{y^{r+1}}\,dy. \]
From the last equality and inequality (2) it follows that
\[ \int_D |f(P)|^r\,dP\int_{f(P)}^\infty \frac{p(y)}{y^r}\,dy \leq \frac{\alpha}{r-\alpha}\int_D M[|f(P)|]\,dP. \]
Finally, for the integral \(I_1\) we obtain the estimate
\[ I_1 \leq \frac{r}{r-\alpha}\int_D M[|f(P)|]\,dP. \tag{12} \]
Let us turn to the second integral. Making the substitution \(y=\beta_f(x)\), integrating by parts, and taking into account that
\[ \lim_{x\to 0} xM[\beta_f(x)] \leq \lim_{x\to 0}\int_0^x M[|f^*(t)|]\,dt =0, \]
we obtain a chain of inequalities:
\[ \begin{aligned} I_2 &=\int_0^\infty \beta_f(y)p(y)\,dy = -\int_0^\infty x\,dM[|\beta_f(x)|] \\ &= -xM[|\beta_f(x)|]\big|_0^\infty + \int_0^\infty M[|\beta_f(x)|]\,dx \leq \int_0^\infty M[|\beta_f(x)|]\,dx \\ &= xM[|\beta_f(x)|]\big|_0^\infty - \int_0^\infty p[|\beta_f(x)|]f^*(x)\,dx + \int_0^\infty \beta_f(x)p[|\beta_f(x)|]\,dx. \end{aligned} \]
From the last equality and inequalities (2), (6), (7), (8) it follows that
\[ (\beta-1)\int_0^\infty M[\beta_f(x)]\,dx \leq \int_0^\infty p[\beta_f(x)]f^*(x)\,dx \leq \|f^*\|_M\|p[\beta_f(x)]\|_N \leq \]
\[ \leq \left[1+\int_0^\infty N\{p[\beta_f(x)]\}\,dx\right]\|f^*\|_M \leq \left[1+\frac{\alpha-1}{\alpha}\int_0^\infty \beta_f(x)p[\beta_f(x)]\,dx\right]\|f^*\|_M \leq \]
\[ \leq 2\left[1+(\alpha-1)\int_0^\infty M[\beta_f(x)]\,dx\right]\|f\|_M^{*}. \]
Now it is easy to verify that on the sphere \(\|f\|_M=k=\dfrac{\beta-1}{4(\alpha-1)}<1\) the inequality
\[ I_2\leq \frac{4}{\beta-1}\,k \tag{13} \]
holds.
Substituting (13), (12) into (11), and then into (10), and using (9), we obtain on the sphere \(\|f\|_M=k\) the estimate
\[ \|\widetilde{f}_\lambda\|_M \leq 1+C_1k\left(\frac{r}{r-\alpha}+\frac{4}{\beta-1}\right). \]
Finally, we shall have
\[ \|K_\lambda\| \leq \frac{1}{k} + C_1\left(\frac{r}{r-\alpha}+\frac{4}{\beta-1}\right). \]
The theorem is proved.
Incidentally, from inequality (13) we have obtained the boundedness of the operator
\[ \beta_f(x)=\frac{1}{x}\int_0^x f(t)\,dt \]
in Orlicz spaces satisfying condition (2) for all \(u\); its norm is
\[ \|\beta\|\leq \frac{4\alpha}{\beta-1}. \]
For \(\mathscr L_p\) \((p>1)\) this result was obtained by Hardy with a more precise estimate of the norm:
\[ \|\beta\|\leq \frac{p}{p-1} \]
([5], p. 77).
The theorem proved admits a generalization to an unbounded domain \(D\) under the following additional conditions: 1) inequality (2) holds for all \(u\); 2) \(\Omega\) depends only on \(\theta\), or the conditions of S. G. Mikhlin’s theorem [3] are satisfied for some \(p>\alpha\).
The theorem is applicable in estimating higher derivatives of elliptic equations, in the theory of one-dimensional singular equations, and in boundary-value problems for analytic functions.
The work was reported at the seminar on nonlinear problems of mechanics at Rostov-on-Don University.
The author expresses his gratitude to the seminar leader I. I. Vorovich, and also to its participants V. I. Yudovich and Yu. P. Krasovskii for useful discussion of the work.
Rostov-on-Don
State University
Received
22 X 1959
CITED LITERATURE
- S. G. Mikhlin, Uspekhi Mat. Nauk, 3, no. 3 (125) (1948).
- A. Calderon, A. Zygmund, Acta Math., 88, no. 1—2, 85 (1952).
- S. G. Mikhlin, DAN, 117, no. 1, 28 (1957).
- M. A. Krasnosel’skii, Ya. B. Rutickii, Convex Functions and Orlicz Spaces, 1953.
- A. Zygmund, Trigonometric Series, 1939.
- Koizumi Simuyu, Proc. Japan Acad., 34, no. 4 (1958).
\[ {}^*\ \|f^*\|_M\leq 2\|f\|_M,\quad \text{since}\quad \left\|\frac{f^*}{\|f\|_M}\right\| = \left\|\left(\frac{f}{\|f\|_M}\right)^*\right\| \leq 1+\int_D M\left(\frac{f}{\|f\|_M}\right)\,dP\leq 2. \]