,.,, L p L p loc,, 3., L p L p loc, Lp L p loc.,.,,.,.,.,, L p, 1 p, L p,. d 1, R d d. E R d. (E, M E, µ)., L p = L p (E). 1 p, E f(x), f(x) p d

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Download ",.,, L p L p loc,, 3., L p L p loc, Lp L p loc.,.,,.,.,.,, L p, 1 p, L p,. d 1, R d d. E R d. (E, M E, µ)., L p = L p (E). 1 p, E f(x), f(x) p d"

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1 1 L p L p loc, L p L p loc, Lp L p loc,., 1 p.,. L p L p., L 1, L 1., L p, L p. L 1., L 1 L 1. L p L p loc L p., L 2 L 2 loc,.,. L p L p loc L p., L p L p loc., L p L p loc 1

2 ,.,, L p L p loc,, 3., L p L p loc, Lp L p loc.,.,,.,.,.,, L p, 1 p, L p,. d 1, R d d. E R d. (E, M E, µ)., L p = L p (E). 1 p, E f(x), f(x) p dx < E 2

3 L p = L p (E)., f(x) = g(x), (a.e. x E), f g., f(x) = g(x), (a.e. x E) L p (E), L p (E), f(x) L p (E),. L p (R d ) L p L p (E) C., f, g L p (E), f + g L p (E)., f L p (E), α C, αf L p (E)., f + g αf, L p (E) C f, g L 2 (E), (f, g) = f(x)g(x)dx E, (f, g) f g., L 2 (E)., f, g, f 1, f 2 L 2 (E), α C, (1) (4) : (1) (f, f) 0., (f, f) = 0 f = 0, (a.e.x E). (2) (f, g) = (g, f). (3) (f 1 + f 2, g) = (f 1, g) + (f 2, g). (4) (αf, g) = α(f, g) , f(x) f(x). 3

4 1.1.1 f, g, g 1, g 2 L 2 (E), α C., (3), (4) : (3) (f, g 1 + g 2 ) = (f, g 1 ) + (f, g 2 ). (4) (f, αg) = α(f, g)., α α. 1 p <, f L p (E), { } 1/p f p = f(x) p dx E. f L p. f p f. L = L (E), f = ess.sup f(x) = inf{α; f(x) α, (a.e. x E)} x E p, L p = L p (E). L p, f, g L p (E), α C, (1) (3) : (1) f 0., f = 0 f = 0, (a.e. x E). (2) f + g f + g, ( ). (3) αf = α f., L 2 f 2 = (f, f) 4

5 ., f, g L 2 (E), (1) (3) : (1) (f, g) f g, ( ). (2) f + g 2 + f g 2 = 2( f 2 + g 2 ), ( ). (3) 4(f, g) = f + g 2 f g 2 + i( f + ig 2 f ig 2 ). 1 p, f, g, h L p (E), ρ(f, g) = g f, f g p, L p (E)., f, g, h L p (E), ρ(f, g) (1) (3) : (1) ρ(f, g) 0., ρ(f, g) = 0 f = g, (a.e. x E). (2) ρ(f, g) = ρ(g, f). (3) ρ(f, g) ρ(f, h) + ρ(h, g)., L p (E)., L p (E) p. L p (E).. 5

6 {f m } L p (E) f L p (E) p, lim m f m f p = 0., {f m } f p. 2, l.i.m.f m (x) = f(x). l.i.m. limit in mean. limit in the mean. {f m } L p (E). lim f l f m p = 0 l, m p, L p (E) p., L p (E) {f m }, L p (E) f., L p (E)., L 2 (E) , p, f m L p (E), (m = 0, 1, 2, ), lim f m f 0 p = 0, {f m } m {f m(k) : k = 1, 2, }, lim f m(k)(x) = f 0 (x), k (a.e. x E) µ(e) <, 1 p, L p (E) L 1 (E). 6

7 1.1.8 p, q 1 < p, q <, 1 p + 1 q = 1. E R d, L p = L p (E)., : L p = (L q ) = (L p ) d 1, 1 < p <, L p = L p (R d ). D = D(R d ) R d C TVS., D L p d 1, 1 < p <, L p = L p (R d ). L p {f n }, f L p., (1), (2), (3) : (1) L p f n f. (2) L p f n f. (3) D L p f n f. 1.2 L p loc, L p loc., 1 p. d 1. R d d. 7

8 1 p <, R d K, K f(x) p dx < f p. R d p L p loc = Lp loc (Rd ). L p loc. 1 p <, f L p loc, R > 0, f(x) p dx < r R., L 1 loc. 1 p <, L p loc {f m} f L p loc, R d K, f m (x) f(x) p dx 0, (m ) K., L p loc L p., L p loc., L loc = L loc (Rd ) R d K, f, K = ess.sup f(x) x K = inf{α; f(x) α, (a.e. x K)} <. L loc f, K = ess.sup f(x) x K 8

9 , L loc {, K ; K R d }. L loc {f m} f L loc, Rd K, f m f, K 0, (m )., L loc L., K loc., 1 p, L p loc L1 loc. 1 p, L p c L p TVS., L p loc FS, L p c DFS, L p loc Lp c., p, q 1 p, 1 q, 1 p + 1 q = 1., (1), (2) : (1) L p loc = (Lq c) = (L p loc ). (2) L q c = (L p ) = (L q c). 9

10 p, D L p c p, L p loc {f m} L p loc f, (1) (3) : (1) L p loc, f m f. (2) L p loc, f m f. (3) D L p loc f m f. 1.3 L p L p, L p. (, ) p L p = L p (, )., 1 p., L p., L p L p., (L p ) y = f(x) (, ) L p., x x y = f(x) y y = f(x + x) f(x) 10

11 = A(x) x + ε(x, x) x., A(x) x x. ε = ε(x, x) x x., y = f(x) (, ) L p (, ), L p, x 0, ε(x, x) 0 lim ε(x, x) = 0 x 0, ε(x, 0) = 0, (x (, ))., L p,, L p. (a, b) p L p = L p (a, b)., f L p (a, b), f(x), (x (a, b)), f(x) = 0, (x / (a, b)), f(x) L p (, )., f(x) L p (a, b) f(x) L p (, ) 1 1, L 2 (a, b) L 2 (, )., L p (a, b) f L p, f L p (, ) L p. 11

12 , y = f(x) (a, b) L p., 1.3.1, L p, y lim x 0 x = lim f(x + x) f(x) = f (x) x 0 x. f (x) y = f(x) L p. L p, f (x) L p (a, b). L p, f (x) (a, b) L p D = D(R) R C. L p = L p (R)., 1 p <., L p f(x) L p., f(x) w-f (x) L 1 loc (w-f, φ) = (f, φ ), (φ D)., p <, f(x) L p. f(x) L p, f(x), L p 12

13 f (x) w-f (x)., f (x) = w-f (x),. (f, φ) = (w-f, φ), (φ D), L p, f(x + x) f(x) lim = f (x) x 0 x., φ D, : ( (f (x), φ(x)) = = lim x 0 = lim x 0 1 = lim x 0 x 1 = lim x 0 x ( = lim x 0 f(x + x) f(x) ), φ(x) x ( f(x + x) f(x) ), φ(x) x 1 { ( ) ( ) } f(x + x), φ(x) f(x), φ(x) x f(x), { ( f(x), φ(x x) ) ( f(x), φ(x) ) } ( f(x), 1 ( ) ) φ(x x) φ(x) x ) φ(x x) φ(x) lim x 0 x = ( f(x), φ (x) ) = ( w-f (x), φ(x) )., f (x) = w-f (x) 13

14 . //, f(x) L p w-f (x) L p < p <, f(x) L p. f(x) w-f (x), w-f (x) L p, f(x) L p, f(x) L p f (x), w-f (x) = f (x) p <, f n (x) L p, (n = 1, 2, 3, ), f, g L p f n f, (n ), f n g, (n ), f L p f = g., d dx , 1 < p <, L p, L p.,, L p L p, L p L p R d p L p = L p (R d )., d 2, 1 p <. 14

15 , L p., L p L p, (R d ) y = f(x) R d L p., x x y = f(x) y y = f(x + x) f(x) = d A i (x) x i + ε(x, x)ρ i=1., A i (x), (i = 1, 2,, d), x x. ε = ε(x, x) x x., y = f(x) R d, L p, R d, L p.,. x 0, ε(x, x) 0 lim ε(x, x) p = 0 x 0, ε(x, 0) = 0, (x R d )., L p,, L p., R d D p L p = L p (D). 1, 15

16 L p (D) L p (R d )., L p (D) f L p, f L p (R d ) L p L p d 2, 1 p <., f(x) L p = L p (R d ) L p, 1 j d, y (τ hej f)(x) f(x) = lim h 0 h L p., 1 j d, {e 1, e 2,, e d } l 2 (d), τ y, (y R d )., y, (1 j d) L p., L p., y, (1 j d). L p y = f(x) L p, y = f(x) L p., f(x) L p L p, f(x) 1 j d, x j L p., f(x) L p, f(x) L p.,.,. 16

17 1.3.4, 1 p <, 1 j d, f(x) L p., f(x) w- f L 1 loc, ( w- f ) (, φ = f, φ ), (φ D)., , 1 p <, 1 j d, f(x) L p. f(x) L p, f(x), L p f w- f.,,. f = w- f, (1 j d) ( f ) (, φ = w- f ), φ, (φ D, 1 j d), f(x) L p w- f, (1 j d) L p. f(x) L p , 1 < p <, 1 j d, f(x) L p., f(x) w- f, w- f L p, f(x) L p, f(x) L p 17

18 f,. w- f = f, (1 j d), p <, f(x) L p. 1 i, j d, (i j), L p 2 f x i 2 f x i,. 2 f x i = 2 f x i, f(x), 2 2 f 2 f w- w-, x i x i., 2 f 2 f = w-, x i x i 2 f 2 f = w- x i x i 2 f 2 f w- = w- x i x i, 2 f x i = 2 f x i 18

19 . // p <, f n (x) L p, (n = 1, 2, 3, ), f, g L p, f n f, (n ), f n g, (n ) f L p f = g., 1 j d., , 1 < p <, L p, L p., L p, L p., 1 < p <, L p L p, L p. 1.4 L p loc L p loc L p loc. (a, b) p L p loc = Lp loc (a, b)., 1 p. 19

20 , (a, b) p L p loc., (L p loc ) y = f(x) (a, b) p., 1 p., x x y = f(x) y y = f(x + x) f(x) = A(x) x + ε(x, x) x., A(x), x, x. ε = ε(x, x) x x., y = f(x) (a, b), L p loc, (a, b), L p loc x 0, ε(x, x) 0.,, a < c < d < b c, d lim q [c, d](ε(x, x)) = lim x 0 x 0. ( d ) 1/p= ε(x, x) p dx 0 ε(x, 0) = 0, (x (a, b)), L p loc,, Lp loc. c 20

21 , y = f(x) (a, b) L p loc., 1.4.1, x (a, b) L p loc, y lim x 0 x = lim f(x + x) f(x) = f (x) x 0 x., f (x) y = f(x) L p loc. L p loc, f (x) L p loc (a, b). L p loc, f (x) (a, b) L p loc D = D(R) R C., L p loc = Lp loc (R)., 1 p., p, f(x) L p loc., f(x) w-f (x) L 1 loc (w-f, φ) = (f, φ ), (φ D)., p, f(x) L p loc. f(x) L p loc, f(x), Lp loc 21

22 f (x) w-f (x)., f (x) = w-f (x), (f, φ) = (w-f, φ), (φ D)., f(x) L p loc w-f (x) L p loc p, f(x) L p loc. f(x) w-f (x), w-f (x) L p loc, f(x) L p loc, f(x) Lp loc f (x), w-f (x) = f (x) p, f n (x) L p loc, (n = 1, 2, 3, ), f, g Lp loc, f n f, (n ), f n g, (n ), f L p loc, f = g., d dx , 1 p, L p loc, Lp loc.,, L p loc Lp loc, Lp loc 22

23 ., L 1 L 1 loc, L1 L 1 loc., L 1, L 1 loc L L p loc D R d., d 2. D p L p loc = Lp loc (D)., 1 p., D p, L p loc., (L p loc ) y = f(x) D p., 1 p., x x y = f(x) y y = f(x + x) f(x) d = A i (x) x i + ε(x, x)ρ i=1., A i (x), (i = 1, 2,, d), x x. ε = ε(x, x) x x., y = f(x) D L p loc, D, L p loc, x 0, ε(x, x) 0 23

24 1 p <,, D K, ( ) 1/p= lim q K(ε(x, x)) = lim ε(x, x) p dx 0 x 0 x 0 K p =, L loc (D). ε(x, 0) = 0, (x D), L p loc,, Lp loc L p loc R d D, L p loc = Lp loc (D)., d 2, 1 p., f(x) L p loc Lp loc, 1 j d, y (τ hej f)(x) f(x) = lim h 0 h L p loc., 1 j d, {e 1, e 2,, e d } l 2 (d), τ y, (y R d )., y, (1 j d) L p x loc., j L p loc., y, (1 j d). 24

25 , , 1 p, 1 j d, f(x) L p loc., f(x) w- f L 1 loc ( w- f ) (, φ = f, φ ), (φ D)., , 1 p, 1 j d, f(x) L p loc. f(x) L p loc, f(x), L p f loc w- f., f = w- f, (1 j d), ( f ) (, φ = w- f ), φ, (φ D, 1 j d)., f(x) L p f loc w- L p x loc j (1 j d). f(x) L p loc , 1 p, f(x) L p loc., 1 j d, f(x) w- f 25

26 , w- f L p loc, f(x) Lp loc, f(x) L p loc. f, w- f = f, (1 j d), p, f(x) L p loc. 1 i, j d, (i j), L p loc 2 f x i 2 f, x i. 2 f x i = 2 f x i, f(x), 2 2 f 2 f w- w-, x i x i., 2 f 2 f = w-, x i x i 2 f 2 f = w- x i x i 2 f 2 f w- = w- x i x i 26

27 ,. // 2 f x i = 2 f x i p, f n (x) L p loc, (n = 1, 2, 3, ), f, g Lp loc, f n f, (n ), f n g, (n ) f L p loc f = g., 1 j d., , 1 p, L p loc, L p loc., L p loc, Lp loc.,, L p loc Lp loc, Lp loc. L 1 L 1 loc, L1, L 1 loc L1. 27

( ) ( ) ( ) i (i = 1, 2,, n) x( ) log(a i x + 1) a i > 0 t i (> 0) T i x i z n z = log(a i x i + 1) i=1 i t i ( ) x i t i (i = 1, 2, n) T n x i T i=1 z = n log(a i x i + 1) i=1 x i t i (i = 1, 2,, n) n

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