The Muckenhoupt-type estimations for the best constants in multidimensional modular inequalities over spherical cones Chang-Pao Chen Department of Applied Mathematics Hsuan Chuang University Hsinchu, Taiwan 30092, R.O.C. Email: cpchen@wmail.hcu.edu.tw and Jin-Wen Lan Department of Mathematics National Tsing Hua University and Dah-Chin Luor Department of Applied Mathematics I-Shou University 1
Notations Σ n − 1 : the unit sphere in R n , E ⊂ R n : a spherical cone, that is, each x ∈ E is of the form x = sτ for some 0 < s < ∞ and some τ ∈ A , A : a given Borel measurable subset of Σ n − 1 , S x = { y ∈ E : y = sτ, 0 < s < | x | , τ ∈ A } , ˜ S x = { y ∈ E : y = sτ, 0 < s ≤ | x | , τ ∈ A } , k ( x , t ) ≥ 0: a locally integrable Borel measurable function defined on E × E , σ : a σ -finite Borel measure on E . 2
Introduction Consider the integral operator � K f ( x ) := k ( x, t ) f ( t ) dσ ( t ) ( x ∈ E ) , ˜ S x (1 . 1) This paper deals with the following modular inequality � 1 � 1 � q � p �� � �� � q p Φ ◦ K f ( x ) dµ ≤ C Φ ◦ f ( x ) dν , E E (1 . 3) where Φ ◦ f ( x ) = Φ( f ( x )), 1 ≤ p, q ≤ ∞ , µ , ν are two σ -finite Borel measures on E , Φ ∈ CV + ( I ) for some open interval I in R . Here CV + ( I ) denotes the class of all nonneg- ative convex functions defined on I . We try to find the smallest constant C such that (1 . 3) holds for some suitable class of f . 3
The problem that we consider in this paper is to estimate � K � ∗ , where � K � ∗ is the “norm” of the operator K : D K ∩ L p Φ ( dν ) �→ L q Φ ( dµ ), D K consists of those f such that K f ( x ) is well-defined for µ a.e. x ∈ E , L p Φ ( dν ) denotes the set of all real-valued Borel measurable f with � f � Φ ,p,ν < ∞ , � p � 1 /p �� � � f � Φ ,p,ν := Φ ◦ f ( x ) dν < ∞ . (1 . 4) E 4
Facts The value of � K � ∗ has been investigated for a long time. (1) It was initiated by the work of Hardy. In [9, Theorem 327], the following inequality was established for 1 < p < ∞ : � ∞ � x � p � p � ∞ � � 1 p f ( x ) p dx 0 f ( t ) dt dx ≤ x p − 1 0 0 ( f ≥ 0) . (1 . 5) � p � It is known that the constant p/ ( p − 1) is the best possible. 5
(2) Hardy’s result was extended in many directions. For instance, Christ and Grafakos (see [3, Theorem 1]) established the following n -dimensional extension of Hardy’s result: � p � 1 � � B ( | x | ) f ( t ) dt dx R n | B ( | x | ) | � p � � p R n f ( x ) p dx ≤ ( f ≥ 0) , (1 . 6) p − 1 where B ( r ) is the closed ball centered at the origin with radius r . Remark : This result is the same as to say that � K � ∗ = p/ ( p − 1), where p = q, Φ( x ) = | x | , k ( x, t ) = 1 / | B ( | x | ) | , E = R n \ { 0 } , dσ = dt, A = Σ n − 1 , dµ = dν = dx. 6
(3) Eq. (1 . 6) has been extended to the case: Φ( x ) = | x | of (1 . 3) for dµ = u ( x ) dx , dν = v ( x ) dx , and the so-called Oinarov kernel (cf. [5, 14, 27, 28] for details). A typical result in this direction (see [28, The- orems 6.2 & 6.3]) says that for 1 < p, q < ∞ and Φ( x ) = | x | , (1 . 3) holds for all 0 ≤ f ∈ L 1 loc ( E ) if and only if max( A 0 , A 1 ) < ∞ . Moreover, � K � ∗ ≈ max( A 0 , A 1 ), where A 0 = � A 0 ( x ) � r,ω 0 , A 1 = � A 1 ( x ) � r,ω 1 , dω 0 = v ( x ) 1 − p ∗ dx, dω 1 = u ( x ) dx and � η ∗ �� p ∗ q ∗ v ( t ) 1 − p ∗ dt A 0 ( x ) = ˜ S x � q � 1 /q �� � k ( z, x ) u ( z ) dz (1 . 7) × , E \ S x 7
� p ∗ � 1 /p ∗ �� � v ( t ) 1 − p ∗ dt A 1 ( x ) = k ( x, t ) ˜ S x � 1 /η �� u ( z ) dz (1 . 8) × . E \ S x Here and in sequel, 1 /r = 1 /q − 1 /η, η = max( p, q ) , and ( · ) ∗ is the conjugate exponent of ( · ). Remark : For k ( x, t ) = 1, it is known that A 1 = A 0 for 1 < p ≤ q < ∞ and A 1 = ( q/p ∗ ) 1 /r A 0 for 1 < q < p < ∞ (by the integration by parts). In this case, � K � ∗ ≈ A 0 . 8
(4) Wedestig’s result only says that C 1 max( A 0 , A 1 ) ≤ � K � ∗ ≤ C 2 max( A 0 , A 1 ) for some constants C 1 and C 2 . However, the best possible values of C 1 and C 2 are still unknown (cf. [14, 28]). In [14, pp.4-5], B. Opic and others consid- ered k ( x, t ) = 1 and established the following Muckenhoupt-type estimates for the case that n = 1 and Φ( x ) = | x | : � η ∗ / ( p ∗ q ∗ ) � 1 /q � 1 + p ∗ � p ∗ + q q ρA 0 ≤ � K � ∗ ≤ A 0 , η η (1 . 9) where 1 ( p ≤ q ) , � 1 /q ∗ � ρ = p ∗ q q 1 /q ( q < p ) . r 9
(5) Later in [8, 27], it was shown that for n ≥ 1 , and Φ( x ) = | x | , A 0 ≤ � K � ∗ ≤ p 1 /q ( p ∗ ) 1 /p ∗ A 0 for 1 < p ≤ q < ∞ and ρ ( q/r ) 1 /q A 0 ≤ � K � ∗ ≤ r 1 /r p 1 /p ( p ∗ ) 1 /q ∗ A 0 for 1 < q < p < ∞ . (6) On the other hand, as indicated in [28, Section 3.4 & Lemma 7.4], there are two other types of estimates instead of the upper bound in (1 . 9). They are p ∗ A PS and � 1 /p ∗ � p − 1 A W := 1 <s<p A W ( s ) inf , (1 . 10) p − s where 1 < p ≤ q < ∞ , � − 1 /p �� v ( t ) 1 − p ∗ dt A PS = sup ˜ S x x ∈ E � q � 1 /q �� �� v ( y ) 1 − p ∗ dy × u ( t ) dt , ˜ ˜ S x S t 10
and � ( s − 1) /p �� v ( t ) 1 − p ∗ dt A W ( s ) = sup ˜ S x x ∈ E � q ( p − s p ) � 1 /q �� �� v ( y ) 1 − p ∗ dy × u ( t ) dt . ˜ E \ S x S t The former was found in Persson-Stepanov [25] and the latter was given in Wedestig [28, The- orem 3.1] (see also [12], [22]). Remark: As claimed in [28, pp.27-29], for some case (e.g. p = 3 , q = 4), the upper bound estimate given in (1 . 9) is better than p ∗ A PS . However, up to now, there is no significant result concerning the comparison problem between the right side of (1 . 9) and (1 . 10). 11
Goal: The purpose of this paper is to establish the following result for those k satisfying � k ( x, t ) dσ ( t ) = 1 ( x ∈ E ) ( ∗ ) ˜ S x Main Result Let 1 ≤ p, q < ∞ , Φ ∈ CV + ( I ), and k ( x, t ) = g ( t ) ψ ( x, t ). Suppose that � dω = 0 for all x ∈ E. ( ∗∗ ) ˜ S x \ S x Then the following assertions hold: ( i ) If A M < ∞ , then � η ∗ / ( p ∗ q ∗ ) � 1 /q � 1 + p ∗ � p ∗ + q q � K � ∗ ≤ A M . η η ( ii ) The condition ( ∗∗ ) is not necessary for the case 1 < p ≤ q < ∞ . For Φ( x ) = | x | , the condition ( ∗ ) can be removed. 12
( iii ) In ( i ), if dµ ( x ) = u ( x ) dx , dν ( x ) = v ( x ) dx , dσ ( x ) = ξ ( x ) dx , and ψ ( x, t ) ≤ ψ ( x, s ) ≤ Dψ ( x, t ) for | t | ≤ | s | ≤ | x | , then for Φ( x ) = | x | , � η ∗ / ( p ∗ q ∗ ) � 1 /q � 1+ p ∗ � p ∗ + q q ρ ∗ A M ≤ � K � ∗ ≤ A M , η η where u ( x ) ≥ 0, v ( x ) ≥ 0, and ξ ( x ) > 0 on E , and D − 1 ( p ≤ q ) , � 1 /q ∗ ρ ∗ = � D − r/q p ∗ q q 1 /q ( q < p ) . r Notations : � p ∗ � g ( t ) A M = � A ( x ) � r,ω , dω ( t ) = dν ( t ) , dν a /dσ η ∗ /q ∗ � � � �� � g ( · ) � � � � A ( x ) = sup ψ ( · , t ) × . � � � � � dν a /dσ � � � L p ∗ (˜ t ∈ ˜ � L q ( E \ S x ,µ ) S x � � S x ,ν ) � 13
Case I : 1 < p = q < ∞ , dσ = dµ = dν = dt , ψ ( x, t ) = 1 /G ( x ), where g ( x 1 ) ≤ g ( x 2 ) ( | x 1 | ≤ | x 2 | ) and � 0 < G ( x ) := g ( t ) dt < ∞ ( x ∈ E ) . ˜ S x The n -dimensional extensions of Levinson’s modular inequality: Then for Φ ∈ CV + ( I ) and Let 1 < p < ∞ . f : E �→ ¯ I , we have �� p � 1 /p �� � � 1 � Φ g ( t ) f ( t ) dt dx G ( x ) ˜ E S x � 1 /p �� E (Φ ◦ f ( x )) p dx ≤ p ∗ . Remark : The result of Christ and Grafakos (see [3, Theorem 1]) corresponds to the case: Φ( x ) = | x | and g ( t ) = 1. 14
Case II : p = q = 1, ψ ( x, t ) = 1 /G ( x ), u ( x ) dσ = dt , dµ ( x ) = χ ˜ | x | dx , and S b � � v ( x ) ρ ( x ) dν ( x ) = χ ˜ | x | + χ E \ ˜ dx ( k = 1 , 2 , · · · ) . S b S b k Generalizations of the Hardy-Knopp-type inequalities: Let � 0 < G ( x ) := g ( t ) dt < ∞ ( x ∈ E ) . ˜ S x Suppose that b ∈ E ∪ {∞} and u : ˜ S b �→ [0 , ∞ ) u ( x ) | x | G ( x ) is locally integrable on ˜ be such that S b . Then for Φ ∈ CV + ( I ) and f : ˜ S b �→ ¯ I , we have � � 1 u ( x ) � � Φ g ( t ) f ( t ) dt | x | dx ˜ G ( x ) ˜ S b S x Φ ◦ f ( x ) v ( x ) � ≤ | x | dx, ˜ S b u ( y ) where v ( x ) = | x | g ( x ) � | y | G ( y ) dy . ˜ S b \ S x 15
Case III : 1 ≤ p ≤ q < ∞ , Φ( x ) = | x | , g ( t ) = e − m | t | , ψ ( x, t ) = e m | x | , and dσ = dµ = dν = | x | 1 − n dx , where m > 0. Extension of Stepanov’s result : � 1 /q q �� � � � � e m ( | x |−| t | ) | t | 1 − n f ( t ) dt � | x | 1 − n dx � � � � E E \ S x � � � 1 /p ∗ +1 /q �� � 1 /p (1 /p ∗ + 1 /q ) | A | � E | f ( x ) | p | x | 1 − n dx ≤ . m Remark : The particular case 1 < p = q < ∞ and n = 1 of the above result reduces to [15, Theorem D83], which was attributed by V. I. Levin and S. B. Steˇ ckin to V. V. Stepanov. 16
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