Approximation properties of the picard singular integral in exponential weighted spaces

Publicacions Matem`atiques, Vol 40 (1996), 233–242. APPROXIMATION PROPERTIES OF THE PICARD SINGULAR INTEGRAL IN EXPONENTIAL WEIGHTED SPACES A. Le´ sniewicz, L. Rempulska and J. Wasiak Abstract In this note we give some direct and inverse approximation theorems for the Picard singular integral in the exponential weighted spaces and some generalized H¨older spaces. 1. Preliminaries 1.1. The Picard singular integral (1) Pr(f;x):= 1 2r+∞ −∞ f(x+t) exp −|t| rdt, x∈R:= (−∞,+∞), r>0 and r→0+, was examined in [1],[2],[4] for functions belonging to the space Lpand the classical H¨older spaces. The purpose of this note is to give some approximation properties of the Picard integral (1) in the exponential weighted spaces Lp,q and some generalized H¨older spaces [5]. 1.2. Let q>0 be a fixed number and let (2) wq(x):=e−q|x|,x∈R. For a fixed 1 ≤p≤∞and q>0 we denote by Lp,q the set of all real-valued functions fdefined on Rfor which the p-th power of wqfis Lebesgue-integrable on Rif 1 ≤p<∞, and wqfis uniformly continuous and bounded on Rif p=∞. Let the norm in Lp,q be given by the formula (3) fp,q ≡f(·)p,q :=        R |wq(x)f(x)|pdx 1 p if 1 ≤p<∞, sup x∈R wq(x)|f(x)|if p=∞. 234 A. Le´ sniewicz, L. Rempulska, J. Wasiak For f∈Lp,q we define the modulus of smoothness of the order 2 (4) ω2(f,Lp,q;t) := sup |h|≤t ∆2 hf(·)p,q,t≥0, where (5) ∆2 hf(x):=f(x+h)+f(x−h)−2f(x),x,h∈R. From (3)-(5) for f∈Lp,q follows (6) f(·+h)p,q ≤eq|h|f(·)p,q,h∈R, (7) 0 = ω2(f,Lp,q;0)≤ω2(f,Lp,q;t1)≤ω2(f,Lp,q;t2)if0<t 1<t 2. Using the indentity (see [3, pp. 25–29]) ∆2 nhf(x)= n  k=1 k∆2 hf(x−(n−k)h)+ n−1  k=1 (n−k)∆2 hf(x+kh), x,h∈R;n=2,3,..., and by (2)-(6) we can prove that ω2(f,Lp,q;nt)≤n2e(n−1)qtω2(f,Lp,q;t) for n=1,2,... and t≥0 and ω2(f,Lp,q;λt)≤(λ+1) 2eλqtω2(f,Lp,q;t) for λ, t ≥0. 1.3. Denote as in [5] by Ω2the set of all functions of order 2 modulus of smoothness type ([6]), i.e., Ω2is the set of all functions ωsatisfying the following conditions (i) ωis defined and continuous on [0,+∞), (ii) ωis increasing and ω(0) = 0, (iii) ω(t)t−2is decreasing on [0,+∞). As in [5], for a given 1 ≤p≤∞,q>0 and ω∈Ω2, we define the generalized H¨older space Lp,q,ω of all functions f∈Lp,q for which the quantity (9) f∗ p,q,ω := sup 0<h≤1∆2 hf(·)p,q ω(h) is finite. The norm in Lp,q,ω is defined by (10) fp,q,ω := fp,q +f∗ p,q,ω. Approximation properties of Picard integral 235 For f∈Lp,q,ω we have (11) ω2(f,Lp,q;t)≤f∗ p,q,ωω(t),0≤t≤1. If ω,µ∈Ω2and the function (12) ϕ(t):=ω(t) µ(t),t>0, is increasing, then for a fixed 1 ≤p≤∞and q>0wehaveLp,q,ω ⊂ Lp,q,µ. It is easy to observe that for every ω∈Ω2there exist two positive constants M1,M2such that (13) M1t2≤ω(t)≤M2t21 t ω(z)z−3dz for all 0 ≤t≤1 2. In this note we shall study the limit properties of the Picard integral (1) for functions belonging to the spaces Lp,q and Lp,q,ω. We first notice that for each r,Pr, as given by (1), is well defined on all functions f∈Lp,q, 1≤p≤∞,q>0, provided ris small enough, i.e., 0 <r<1 q. It is then a linear positive operator. In Section 2 we shall prove that, for a given 1≤p≤∞,q>0 and 0 <r<1 q,Pris an operator from Lp,q into Lp,q. Moreover, we shall prove that, for a given 1 ≤p≤∞,q>0, ω∈Ω2 and 0 <r<1 q,Pris an operator from Lp,q,ω into Lp,q,ω. 2. Auxiliary results In this part we shall give some fundamental properties of the Picard integral Prin the spaces Lp,q and Lp,q,ω. It is easy to verify that holds. Lemma 1. For k=0,1,2,... and y>0we have +∞ 0 tkexp(−yt)dt =k!y−k−1. In what follows, for a given q>0, we shall denote by r0≡r0(q)a fixed number such that (14) 0 <r 0<1 q. 236 A. Le´ sniewicz, L. Rempulska, J. Wasiak Lemma 2. For every fixed 1≤p≤∞and q>0, the Picard integral Pris an operator from Lp,q into Lp,q provided that 0<r<1 q. Moreover, (15) Pr(f;·)p,q ≤1 1−r0qfp,q for 0<r≤r0, where r0is given by (14). Proof: We shall prove only (15). If f∈L∞,q,q>0, then by (1)-(3) we have wq(x)|Pr(f;x)|≤f∞,qe−q|x|(2r)−1R eq|x+t|− |t| rdt ≤f∞,qr−1+∞ 0 et(q−1 r)dt, x ∈R, and further for 0 <r≤r0<1 qwe get by Lemma 1 Pr(f;·)∞,q ≤f∞,q 1 1−rq ≤1 1−r0qf∞,q. Similarly, if f∈Lp,q,1≤p<∞and q>0, then by (1)-(3) and the generalized Minkowski inequality we get Pr(f;·)p,q ≤fp,q(2r)−1R e|t|(q−1 r)dt, which by Lemma 1 implies (15) for r∈(0,r 0]. Thus the proof of (15) is completed. Lemma 3. Let Lp,q,ω be a given H¨older space (1 ≤p≤∞,q>0, ω∈Ω2)and let r0be given by (14). Then for every f∈Lp,q,ω and r∈(0,r 0]holds (16) Pr(f;·)p,q,ω ≤1 1−r0qfp,q,ω, which proves that Pr,0<r<1 q, is an operator from Lp,q,ω into Lp,q,ω. Proof: From (1) and (5) follows (17) ∆2 hPr(f;x)=Pr(∆2 hf;x) for all x, h ∈Rand 0 <r<1 q. Hence and by (9) and (15) we have (18) Pr(f;·)∗ p,q,ω := sup 0<h≤1Pr∆2 hf;·p,q ω(h) ≤1 1−r0qsup 0<h≤1∆2 hf;(·)p,q ω(h)=1 1−r0qf∗ p,q,ω for all 0 <r≤r0<1 q. Combining (15), (18) and (10), we obtain (16) and we complete the proof. Approximation properties of Picard integral 237 Lemma 4. Suppose that f∈Lp,q with some 1≤p≤∞,q>0, and r0is given by (14). Then (19) Pr1(Pr2(f); x)=Pr2(Pr1(f); x)for x∈R,r 1,r 2∈0,1 q. Moreover, for every fixed r∈0,1 qthe function Pr(f;·)has derivatives of all orders belonging also to Lp,q and (20) P(n) r(f;·)p,q ≤(1 −r0q)−1r−nfp,q for all n=1,2,... and r∈(0,r 0]. Proof: For f∈Lp,q,n=1,2,... and 0 <r<1 qwe get from (1) P(n) r(f;x)=r−nPr(f;x),x∈R. This fact and Lemma 2 imply (20). The equality (19) we immediately obtain from (1) and Lemma 2. Lemma 5. Suppose that the assumption of Lemma 4 is satisfied. Then (21) ∆2 hPr(f;·)p,q ≤1 1−r0qfp,qr−2h2eq|h| for all r∈(0,r 0]and h∈R. Proof: By Lemma 4 and (5) for x,h∈Rand 0 <r<1 qwe can write ∆2 hPr(f;x)=h 2 −h 2h 2 −h 2 P r(f;x+t1+t2)dt1dt2. Now arguing as in the proof of Lemma 2 we get ∆2 hPr(f;x)p,q =P r(f;·)p,q h 2 −h 2h 2 −h 2 eq|t1+t2|dt1dt2 ≤P r(f;·)p,qh2eq|h|, which by (20) yields the desired inequality (21). 238 A. Le´ sniewicz, L. Rempulska, J. Wasiak 3. Approximation theorems 3.1. First we shall prove a direct approximation theorem for functions belonging to Lp,q. Theorem 1. If f∈Lp,q with some 1≤p≤∞and q>0, and if r0 is given by (14), then (22) Pr(f;·)−f(·)p,q ≤5 2(1 −r0q)−3ω2(f,Lp,q;r) for all r∈(0,r 0]. Proof: From (1), Lemma 1 and (5) follows Pr(f;x)−f(x)= 1 2r+∞ 0 (∆2 tf(x))e−t rdt for x∈Rand r∈0,1 q. Further, by some calculations as in the proof of Lemma 2 and by (4) and (8), we get Pr(f)−fp,q ≤1 2r+∞ 0 ∆2 tf(·)p,qe−t rdt≤1 2r+∞ 0 ω2(f,Lp,q;t)e−t rdt ≤ω2(f,Lp,q;r)1 2r+∞ 0t r+1 2 e−(1 r−q)tdt. Now using Lemma 1, we easily obtain (22). Theorem 1 and (11) imply the following Corollary 1. If f∈Lp,q,ω with some fixed 1≤p≤∞,q>0and ω∈Ω2, then Pr(f;·)−f(·)p,q ≤5 2(1 −r0q)−3f∗ p,q,ωω(r) for all r∈(0,1] ∩(0,r 0], where 0<r 0<1 q. In particular, if ω(t)=tαwith some 0<α≤2, then Pr(f;·)−f(·)p,q =O(rα)as r→0+. 3.2. Using the Bernstein method ([6, p. 345]) we shall prove an inverse approximation theorem for the Picard integral Pr. Approximation properties of Picard integral 239 Theorem 2. Suppose that f∈Lp,q with some 1≤p≤∞,q>0and (23) Pr(f;·)−f(·)p,q ≤ω(r)for (0,r 0], where ωis a given function belonging to Ω2and 0<r 0<1 q. Then there exists a positive constant Mdepending only on p,q,ω,r0and fp,q such that (24) ω2(f,Lp,q;t)≤Mt21 t ω(z)z−3dz for all t∈0,1 2∩0,1 q. Proof: Choosing two natural numbers n0and nsuch that 0 <2−n< 2−n0<1 q, we can write f(x)=P2−n0(f;x)+ n−1  k=n0 {P2−k−1(f;x)−P2−k(f;x)} +f(x)−P2−n(f;x),x∈R, and further by (5) ∆2 hf(x)=∆ 2 hP2−n0(f;x)+ n−1  k=n0 ∆2 h{P2−k−1(f;x)−P2−k(f;x)} +∆ 2 h{f(x)−P2−n(f;x)},x,h∈R. Using Lemma 5, we get ∆2 hP2−n0(f;·)p,q ≤(1 −r0q)−1fp,q22n0h2eq|h|. From (1), (5) and (19) follows ∆2 h{P2−k−1(f;x)−P2−k(f;x)} =∆ 2 hP2−k−1(f−P2−k(f); x)+∆ 2 hP2−k(P2−k−1(f)−f;x). Hence using Lemma 5 and (23), we obtain ∆2 h{P2−k−1(f;·)−P2−k(f;·)}p,q ≤∆2 hP2−k−1(f−P2−k(f); ·)p,q +∆2 hP2−k(f−P2−k−1(f); ·)p,q ≤(1 −r0q)−1eq|h|h2{22k+2f−P2−k(f)p,q +2 2kf−P2−k−1(f)p,q} ≤(1 −r0q)−1eq|h|h222k+2ω(2−k) for h∈R. 240 A. Le´ sniewicz, L. Rempulska, J. Wasiak By (5), (6) and (23) we have for h∈R ∆2 h{f(·)−P2−n(f;·)}p,q ≤f(·+h)−P2−n(f;·+h)p,q +f(·−h)−P2−n(f;·−h)p,q +2f(·)−P2−n(f;·)p,q ≤2eq|h|+1 ω(2−n). Consequently (25) ∆2 hf(·)p,q ≤eq|h|4n0(1 −r0q)−1h2fp,q +(1 −r0q)−1h2 n−1  k=n0 22k+3ω(2−k)+4ω(2−n) for all h∈R. Now let t∈0,1 2∩0,1 q,|h|≤t,n0<nand let nbe a natural number such that 2−n≤t<2−n+1. Then we get form (25) ω2(f,Lp,q;t)≤M1t2+t2 n−1  k=n0 22k+3ω(2−k)+ω(t), where M1=e4n0(1 −r0q)−1fp,q +(1−r0q)−1+4 . Since ω(s)s−2 is decreasing for s>0, we obtain n−1  k=n0 22kω(2−k)≤n n0 22sω(2−s)ds ≤4 ln 21 t ω(z) z3dz. Collecting the above estimates and using (13), we obtain the desired inequality (24). From Theorem 2 we derive the following Corollary 2. If the assumptions of Theorem 2 are satisfied and ω(t)≤Mt α,t>0, with some m>0and 0<α≤2, then ω2(f,Lp,q;t)=O(tα)if 0<α<2, Ot2|ln t|if α=2,as t→0+. 3.3. Now we shall give an analogue of Theorem 1 in the H¨older norm. Approximation properties of Picard integral 241 Theorem 3. Suppose that ω,µ∈Ω2and the function ϕdefined by (12) is increasing. If f∈Lp,q,ω with some 1≤p≤∞and q>0and if r0is given by (14), then there exists a positive constant Mdepending only on p,q,r0,µsuch that (26) Pr(f;·)−f(·)p,q,µ ≤Mf∗ p,q,ωϕ(r) for all r∈(0,r 0]∩(0,1]. Proof: The assumptions on ω,µand ϕimply that f,Pr(f)∈Lp,q,µ if r∈0,1 q. Let rbe a fixed point in (0,r 0]∩(0,1] and let A=(0,r] and B=(r, 1]. Then by (9) and (10) we have Pr(f;·)p,q,µ ≤Pr(f;·)−f(·)p,q + sup h∈A ∆2 h[Pr(f;·)−f(·)] p,q µ(h) + sup h∈B ∆2 h[Pr(f;·)−f(·)] p,q µ(h):= Y1+Y2+Y3. Using Corollary 1 and by the properties (i), (ii) of functions belonging to Ω2,weget Y1≤5 2(1 −r0q)−3f∗ p,q,ωω(r)≤5 2(1 −r0q)−3µ(r0)f∗ p,q,ωϕ(r). By (6) and Corollary 1, Y3=1 µ(r)sup h∈B 4eqhPr(f;·)−f(·)p,q ≤10eq(1 −r0q)−3f∗ p,q,ωϕ(r). In view of (5), (17) and (15) we have Y2≤sup h∈A ∆2 hPr(f;·)p,q +∆2 hf(·)p,q µ(h)≤sup h∈A Pr(∆2 hf;·)p,q +∆2 hf(·)p,q µ(h) ≤1 1−r0q+1 sup h∈A ϕ(h)∆2 hfp,q ω(h)≤1 1−r0q+1 ϕ(r)f∗ p,q,ω. Summarizing, we obtain (26). From Theorem 3 follows Corollary 3. Let ω(t)=tα,µ(t)=tβfor t>0and let 0<β<α≤ 2.Iff∈Lp,q,ω with some 1≤p≤∞and q>0, then Pr(f;·)−f(·)p,q,µ =O(rα−β)as r→0+.