Hong Rae Cho and Ern Gun Kwon. dv q

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1 J Korean Math Soc 4 (23), No 3, pp SOBOLEV-TYPE EMBEING THEOREMS FOR HARMONIC AN HOLOMORPHIC SOBOLEV SPACES Hong Rae Cho and Ern Gun Kwon Abstract In this paper we consider Sobolev-type embedding theorems for harmonic and holomorphic Sobolev spaces on a bounded domain with C 2 boundary Introduction and statement of results Let be a bounded domain in R N with C 2 boundary For x let δ (x) denote the distance from x to For < p, q < let h p,q be the L p -norms with respect the weighted measures dv q (x) = δ (x) q dv (x) For a function h we define a functional m+σ,p,q, where m is a non-negative integer, < p, q <, and σ, as follows: /p m h m,p,q := j h p dv q if σ =, h m+σ,p,q := j= { h p m,p,q + m+ h p δ ( σ)p dv q } /p We define harmonic Sobolev spaces H m+σ,p,q () by H m+σ,p,q () = {h harmonic on : h m+σ,p,q < } Received October, 22 2 Mathematics Subject Classification: 32A36, 46E35 Key words and phrases: Sobolev-type embedding, harmonic Sobolev space, holomorphic Sobolev space The first author was supported by the Korea Research Foundation Grant (KRF- 2-5-P8) and the second author was supported by grant No R from the Basic Research Program of the Korea Science & Engineering Foundation

2 436 Hong Rae Cho and Ern Gun Kwon Theorem Let be a bounded domain in R N with C 2 boundary Let p <, q i >, s i (i =, 2) with q q = (s s )p Then we have H s,p,q () = H s,p,q () Let be a bounded domain in C n with C 2 boundary Let m be a non-negative integer, < p, q <, and σ We define holomorphic Sobolev spaces A m+σ,p,q () by A m+σ,p,q () = {f holomorphic on : f m+σ,p,q < } Theorem 2 Let < p < p <, q i >, s i (i =, ) with (n + q )/p (n + q )/p = s s Then we have A s,p,q () A s,p,q () Remark 3 (i) According to Theorem, a derivative is compensated for by a factor δ, so that the norms s,p,q and s,p,q are equivalent if q q equals the difference in the number of derivatives, which is (s s )p Beatrous [2] proved this result for the holomorphic Sobolev spaces on a strongly pseudoconvex domain (ii) In Theorem 2, if s = and q = q =, then we have W s,p () O() L p () O(), p = p s n + This is the Sobolev embedding theorem for holomorphic Sobolev spaces In this holomorphic Sobolev embedding theorem the dimension 2n is replaced by n + ; the complex tangent derivatives are counted for one half (iii) Theorem 2 has been proved by Beatrous-Burbea [3] for the unit ball and by Beatrous [2] for the strongly preudoconvex domain The key point is the reproducing kernel with right estimate matching quasimetric on As usual we study the behavior of holomorphic functions in terms of the basic invariant objects attached to the domain; the Bergman kernel and its metric, the Szegö kernel, and the Poisson-Szegö kernel, since all these would naturally be taken into account the simple geometric considerations However in general domains much enough is known about these domain functions and so we must use a different approach For Sobolev norm estimates we replace the role of the reproducing kernel by the mean value theorem and Hardy s inequalities In Section 4 we observe that the assumption of C 2 -smoothness of the boundary of is an essential condition for the Sobolev-type embedding of Theorem 2 We give a counter-example of a convex domain with

3 Sobolev-type embedding theorems 437 C,λ smooth boundary for < λ < which does not satisfy our embedding results Here a C,λ -function means that first derivatives of the function are Lipschitz continuous of order λ The counter-example shows that even a little loss of derivatives of the boundary is not permitted for the sharp embedding results 2 Harmonic Sobolev spaces Throughout this section h will denote a harmonic function on Lemma 2 Let K Let α Z N + and < p <, < q < Then we have sup α h C h p,q, K where the constant C is independent of h Proof Let x K Since h is harmonic, it follows that (see [6]) ( /p α (2) h(x) δ (x) N/p+ α h(y) p dv (y)) B(x,δ (x)/2) We have (22) δ (x)/2 δ (y) 2δ (x) for y B(x, δ (x)/2) By (22), the right-hand side of (2) is bounded by dist(k, ) N/p α (q )/p h p,q By the maximum principle, we get the result Lemma 22 Let s, p <, < q <, and let k be a non-negative integer such that (k s)p + q > Then there is a positive constant C, depending on s, p, q, and k, such that k+ h p δ (k+ s)p dv q C dv q Proof We use a Whitney decomposition of (see []) Let ɛ > be sufficiently small With ɛ fixed, there exists a sequence {x j } in, and a positive integer M, where M depends only on, such that (23) (24) (25) B(x j, δ (x j )) are pairwise disjoint, j B(x j, δ (x j )/2) =, each point of lies in at most M of the sets B(x j, δ (x j ))

4 438 Hong Rae Cho and Ern Gun Kwon Let x B(x j, δ (x j )/2) Then B(x, δ (x)/4) B(x j, δ (x j )) Since k u is harmonic, it follows that δ (x) k h(x) δ (x) N k h(y) dv (y) By Hölder s inequality, we have (26) δ (x) p k+ h(x) p δ (x) N B(x,δ (x)/4) B(x,δ (x)/4) k h(y) p dv (y) Since δ (x)/2 δ (y) 2δ (x) for y B(x, δ (x)/4), it follows from (26) that k+ h(x) p δ (k+ s)p+q δ (x) N Thus it follows that (27) B(x j,δ (x j )/2) B(x,δ (x)/4) k+ h(x) p δ (k+ s)p dv q B(x,δ (x)/4) B(x j,δ (x j )) By (24) and (25), and (27), we have k+ h(x) p δ (k+ s)p dv q j j M k h(y) p δ (y) (k s)p dv q (y) k h(y) p δ (y) (k s)p dv q (y) k h(y) p δ (y) (k s)p dv q (y) B(x j,δ (x j )/2) B(x j,δ (x j )) dv q k+ h p δ (k+ s)p dv q dv q Lemma 23 Let s, p, q, and k be the same as in Lemma 22 There is a compact subset K of such that dv q k+ h p δ (k+ s)p dv q + sup k h p \K \K K

5 Sobolev-type embedding theorems 439 Proof Since is C 2, there is a C vector field ν such that ν(y) is the outward unit normal vector at y For δ > we let δ = {y tν(y) : y, t > δ} Then there exists a number δ > such that the map Φ(y, t) = y tν(y) is a C diffeomorphism of (, δ ) onto \ δ Thus we have \ δ Since k h(y tν(y)) = = dv q δ t δ t δ k h(y tν(y)) p dσ(y)t (k s)p+q dt d dτ k h(y τν(y))dτ + k h(y δ ν(y)) k+ h(y τν(y)) ν(y)dτ + k (y δ ν(y)), it follows that \ dv q δ δ ( δ p k+ h(y τν(y)) dτd) dσ(y)t (k s)p+q dt δ + t k h(y δ ν(y)) p dσ(y)t (k s)p+q dt By Hardy s inequality, we have δ ( δ p k+ h(y τν(y)) dτ) t (k s)p+q dtdσ t Thus it follows that δ \ δ k+ h(y τν(y)) p τ p+(k s)p+q dτdσ dv q By the maximum principle, we get the result k+ h p δ (k+ s)p \ δ + δ (k s)p+q sup δ k h p Proof of Theorem Let m i = [s i ] and σ i = s i m i By symmetry, it is enough to prove the inequality h s,p,q h s,p,q for dv q

6 44 Hong Rae Cho and Ern Gun Kwon h H p,q,s () By Lemmas 2, 22, and 23, we have h p s i,p,q i h p p,q i + mi+ h p δ ( σ i)p dv qi First we assume that s s By Lemma 2 and Lemma 23, it follows that h p p,q m+ h p δ (m +)p dv q + h p p,q (28) m+ h p δ ( σ )p dv q + h p p,q h p s,p,q By the similar method as (28), it follows that m+ h p δ ( σ )p dv q m+ h p δ ( σ )p (29) dv q + h p p,q By (28) and (29), we have h s,p,q h s,p,q for s s Now we assume that s s By Lemma 22, it follows that m+ h p δ ( σ )p dv q m+ h p δ ( σ )p dv q Clearly, h p,q h p,q Thus it follows that h s,p,q h s,p,q for s s 3 Holomorphic Sobolev spaces Theorem 3 Let < p, q <, and s Let f A s,p,q () Let m be a non-negative integer with m s Then we have (3) sup{δ (z) m (s (n+q)/p) m f(z) : z } f s,p,q Proof For p sufficiently near, we translate and rotate the coordinate system so that z(p ) = and the Im z axis is perpendicular to Let B ɛ (p ) denote the non-isotropic ball B ɛ (p ) = { z 2 n (ɛδ (p )) 2 + z j 2 ɛδ (p ) < 2 }

7 Sobolev-type embedding theorems 44 Since is C 2, it follows that there is an ɛ > such that for p sufficiently near and z B ɛ (p ) we have z and (32) δ (p ) 2 δ (z) 2δ (p ) (see []) Since the plurisubharmonicity of m f p is invariant by the affinity ( ) z (z, z 2,, z n ) ɛ δ (p o ), z 2 ɛ δ (p ),, z n, ɛ δ (p ) it follows that (33) m f(p ) p m f(z) p dv (z) Vol(B ɛ (p )) B ɛ (p ) By (32), the right-hand side of (33) is bounded by (34) δ (p ) n+q+(m s)p m f δ (m s)p dv q Thus we get the result of the case m = s Now if m > s, then by Lemma 22, the right-hand side of (33) is bounded by δ (p ) n+q+(m s)p [s]+ f p δ ([s]+ s)p dv q Thus we have m f(p ) δ (p ) m+(s (n+q)/p) f s,p,q By Hardy-Littlewood lemma and Theorem 3, we get the following results Corollary 32 Let < p, q < Then we have (i) A s,p,q () Λ s (n+q)/p (), if s > (n + q)/p (ii) A s,p,q () BMOA(), if s = (n + q)/p Lemma 33 Let < p p <, q i > (i =, 2) with (n + q )/p = (n + q )/p For s we have A s,p,q () A s,p,q () and the inclusion is continuous

8 442 Hong Rae Cho and Ern Gun Kwon Proof We will show that f s,p,q f s,p,q for f A s,p,q () Let s = [s] + σ For j [s] we have j f p dv q = j f p j f p p δ q δ(n+q )(p /p ) dv (35) ( ) ( j f p dv q sup δ (n+q )/p f ) j p p By (35), it follows that (36) j f p,q j f p /p p,q (sup δ (n+q )/p j f ) p /p j f p,q + sup δ (n+q )/p j f By (3), the right-hand side of (36) is bounded by j f p q Thus it follows that (37) f [s],p,q f [s],p,q (38) Now it follows that [s]+ f p δ ( σ)p dv q By (3), it follows that [s]+ f p δ ( σ)p dv q ( sup δ σ+(n+q )/p f ) [s]+ p p (39) sup δ σ+(n+q )/p [s]+ f f s,p,q By (37), (38), and (39), we get the result Proof of Theorem 2 By Lemma 33, we have (3) A s,p,q () A s,p,q (), where (n + q )/p = (n + q)/p Since q q = (s s )p, by Theorem, we have (3) A s,p,q () A s,p,q () By (3) and (3), we get the required result

9 4 A counter-example Sobolev-type embedding theorems 443 In this section we observe that the assumption of C 2 -smoothness of the boundary of is an essential condition for the sharp embedding of Theorem 2 Example 4 We consider the domain defined by = {(z, z 2 ) C 2 : z 2 + z 2 +λ < }, where < λ < We can see that is a bounded convex domain with C,λ boundary, but it has no C 2 boundary In [4] we have compared the growth rate of the functions in A p,q () between this domain and the bounded domain with C 2 boundary Let < p < p <, q i >, s i (i =, ) with ( + q )/p ( + q )/p = s s Let f(z, z 2 ) be some branch of ( z ) d on, where ( + q + 2/( + λ))/p s < d < ( + q + 2/( + λ))/p s We prove that f A s,p,q (), f / A s,p,q () The two facts above imply that A s,p,q () cannot be embedded into A s,p,q () Since is a Lipschitz domain, we have (4) z 2 z 2 +λ δ (z, z 2 ) for (z, z 2 ) (see Lemma 2 in [6], Section 32 of Chapter VI) We have [si]+ f z d+[s i]+ Set r(z ) = ( z 2 ) /(+λ) By (4), it follows that [s]+ f p δ ( σ )p dv q da(z ) z < z (d+[s ]+)p ( z 2 z 2 +λ ) ( σ )p +q da(z 2 ) z 2 <r(z ) We estimate the integral I(z ) = ( z 2 z 2 +λ ) ( σ )p +q da(z 2 ) z 2 <r(z )

10 444 Hong Rae Cho and Ern Gun Kwon By the polar coordinates, we have Note that I(z ) r(z ) z 2 ( z 2 r +λ ) ( σ )p +q rdr ( z 2 s) ( σ )p +q s 2/(+λ) ds ( z 2 ) 2/(+λ)+( σ )p +q ( τ) ( σ )p +q τ 2/(+λ) dτ ( τ) ( σ )p +q τ 2/(+λ) dτ = B ( ) 2 + λ, ( σ )p + q, where B(, ) is the beta function Hence we have [s]+ f p δ ( σ )p dv q ( z 2 ) 2/(+λ)+( σ )p +q z < z (d+[s da(z ) ]+)p ( z 2 ) 2/(+λ)+( σ )p +q = lim r rz (d+[s da(z )da(z ) ]+)p z < lim r ( r 2 ) (d+s )p q 2/(+λ), since (d+s )p q 2/(+λ) < (see [5]) Hence f A s,p,q () By the similar calculation as above, we get [s]+ f p δ ( σ )p dv q lim r ( r 2 ) (d+s )p q 2/(+λ) =, since (d + s )p q 2/( + λ) > Hence f / A s,p,q () References [] F Beatrous, L p estimates for extensions of holomorphic functions, Michigan Math J 32 (985), [2], Estimates for derivatives of holomorphic functions in pseudoconvex domains, Math Z 9 (986), 9 6 [3] F Beatrous and J Burbea, Holomorphic Sobolev spaces on the ball, issertationes Math 256 (989), 57 [4] H R Cho and E G Kwon, Growth rate of the functions in Bergman type spaces, submitted

11 Sobolev-type embedding theorems 445 [5] W Rudin, Function theory in the unit ball of C n, Spinger-Verlag, New York, 98 [6] E M Stein, Singular integrals and differentiability properties of functions, Princeton Univ Press, Princeton, NJ, 97 Hong Rae Cho epartment of Mathematics Pusan National University Pusan , Korea chohr@pusanackr Ern Gun Kwon epartment of Mathematics Education Andong National University Andong , Korea egkwon@andongackr